THE MAIN MENU

The Statistics Module Main Menu is similar to the Graphics Module Main
Menu.  The screen is divided into several parts.  Statistical tests and
functions are listed at the bottom of the screen.  The five Statistical
Functions include: (1) individual summary information, sample depth
information, and running fire events; (2) analyses of the seasonality of
fires; (3) fire interval analyses; (4) tests for changes in temporal
characteristics of the fire regime; and (5) tests for spatial differences
between two fire regimes.  Note the status line at the bottom of the menu
which provides hot key information.  From within any menu, the user can:
(1) run the currently selected set of analyses and view the resulting
output file by tying "G" for Go; (2) view the last output file created
during the analyses by typing "V"; or (3) exit temporarily to the
operating system by typing "D."

ENTERING A DATA FILE NAME

Before any graph can be created, the user must supply a data file
containing the fire history information to be analyzed.  After selecting
"E" to enter a data file name, a dialog box opens that asks the user to
supply a file name, such as "PET.DAT".  Remember that the full DOS path
must be typed in as well preceding the file name if the user has not
specified the path in the Workspace environment.  The program will then
search the directory for the file and, if found, will read in all data,
usually in a fraction of a second.  To indicate that data are being read,
the years are printed in the dialog box as each line in the data file is
read.  After all data have been read, the file name and DOS path are
printed beside the line, along with the number of samples in the data
file.  If the data file is not found, the program will alert the user and
return to the Main Menu.

SPECIFYING AN OUTPUT FILE NAME

After data have been read in, the program automatically will assign a
file name to the output file that will contain the results from the
statistical analyses, such as "PET01.OUT".  The program will always take
the left-hand portion of the data file name, append "01" to it, then
append ".OUT" to indicate that this is an output file.  The user can
always change the file name by selecting "O" from the Main Menu.
Remember that the output file will be written to the directory path
specified in the Workspace environment, or will be written to the current
directory in which the user is logged if no Workspace path has been
specified.  After entering a new output file name, the name will be
displayed beside the second line.

The main advantage to using this function is that the user can develop an
unlimited number of output files for the same data set simply by using
different file names for the output file.  For example, the user can
create an output file containing results from analyses over the full
sample size represented in the data file.  The user can then create
another output file using a different name that would contain results
from a more specific set of analyses about the data file.  For example,
the user may wish to run a set of tests using only a subset of years from
the original data file.  The subsequent output file names can be
numbered, such as "PET02.OUT," or the file names can be more descriptive
regarding the type of results contained in that particular set of
analyses, such as "FI.PET" for fire interval analyses on data set
PET.DAT.

VIEWING AN OUTPUT FILE

Once an analysis has been conducted, the FHX2 software will immediately
read the output file produced from the analyses, and display its contents
on screen.  The results from the last analysis conducted can be viewed at
any time by simply selecting "V" from anywhere within any menu currently
open in the Statistics Module.  Depending on the number of analyses
selected, the output file produced can be quite large.  The user can
navigate through this output file by using the available commands at the
top of the screen once the output has been displayed.  Note that the name
of the output file currently being viewed is displayed at the bottom left
hand corner of the screen.

To advance the output one line at a time, select "P" to move the output
file up one line, and "N" to move it down one line.  To advance the
output one page at a time, select "D" to move the output down one page,
and "U" to move the output up one page.  To advance immediately to the
end of the output file, select "E" (for "end").  To advance immediately
to the beginning of the output file, select "B" (for "beginning").  To
print the output file, select "P" from this menu.  Note that the output
file is automatically sent to the LPT1 port.  No other port options are
available unless the user reassigns the port designations.  To quit and
re-enter the Statistics Module menu, select "Q."

SETTING THE RANGE TO ANALYZE

As in the Graphics Module, the user can select only a certain subset of
the entire range of years to analyze.  Selecting a subset of years to
analyze is crucial to the overall analyses, because the user should
ensure that certain analyses are conducted over a set of years that has
an adequate sample depth.  Determining what an adequate sample depth is
depends on the site itself, the number of samples collected at the site,
the quality of the samples collected, the average number of fire events
recorded per sample, and the objectives of the particular research.
Despite the advances made in fire history research, determining what
constitutes an adequate sample depth is largely a subjective decision by
the researcher.  However, some guidelines can be developed to help the
researcher determine when to begin and end the analyses.  For example, an
adequate sample depth may begin in the year in which at least five
samples record a particular fire event, or in the year in which 50% of
all samples recorded a fire scar.  The user may select to end the
analyses during a period when a marked change in the fire regime occurs,
such as around 1880 when, in the western United States, the fire regime
becomes markedly changed due to various human-related factors.  The user
should carefully inspect the Composite Axis to help determine when to
begin the analyses (see the documentation for the Composite Axis in the
Graphics Module for more information).

Selecting "R" from the Main Menu opens a new window in which the user can
change the range of years to be analyzed.  To change the beginning or
ending years, type "F" (for "First" year) or "L" (for "Last" year) and
input the new beginning or ending years in the dialog box.  The user can
select any subset range of years between the beginning and ending years
contained within the data file.  The default (original) beginning and
ending years can always be returned by selecting "Reset Range" from this
menu.

SELECTING SERIES TO ANALYZE

By default, the program will analyze all series contained in a data file,
indicated on the line beside this function (e.g. "Series: 1 to 40"). 
Occasionally the user will want to analyze a particular subset of the
data file.  For example, the data file may actually contain two sites, a
high elevation site with 15 samples and a low elevation site with 20
samples, that were combined into one data file using the Merge Function
in the Data Entry Module.  The user may wish to only analyze these 15
samples from the higher elevation site.  To select which subset of
samples to analyze, select "P" from this menu.  A new menu will open with
the following functions: 1) First series to analyze; 2) Last series to
analyze; 3) Reset series; and, 4) Show series.  The first and last series
to be analyzed will be displayed to the right of the first two lines.

To change the first and last series to be analyzed, simply select either
of the first two functions and enter the appropriate sample number. 
Remember that the sample number is simply the column that sample occupies
in the data file.  If the user is not sure which columns contain the
subset of samples to be analyzed, select "Show Series" from this menu,
and the series ID's will appear along with the column number which they
occupy in the data file.  Note the column numbers that contain the subset
of samples and enter these columns in the dialog boxes that open for the
first two functions.  If at any point the user wishes to reset the series
back to the default of all series, select "Reset Series," and the
original first and last series to be analyzed will appear to the right of
the first two lines.

SUMMARY INFORMATION

The first set of analyses that should always be conducted is to generate
summary information about the individual samples, and about the site in
general.  The information output from these analyses should always be
printed in a hardcopy format and archived, perhaps in a binder. 
Selecting "U" from the Main Menu will open a new window containing
several functions for outputting summary information.  These functions
include: (1) individual summaries; (2) sample depth information; and (3)
fire frequency information for various durations.  Note that all of these
functions are simple toggles that switch between "Yes" and "No."  In the
following discussions, it will be assumed that the user has selected to
run and view the selected analysis being discussed by selecting "G" from
the Statistics Module Main Menu.

Individual Summaries
Selecting "I" from this menu will output all information about each
individual sample to the specified output file.  This function should
always be performed prior to conducting any additional tests.  Once the
information has been output, the user should check the output and ensure
that all information entered about each individual sample is correct by
comparing the information to the fire history sample sheets that contain
the original information.  This is necessary because errors during data
input are bound to happen, and any seemingly minor error may change the
character of the fire information from any one sample.  Once all
information has been verified, the printed output from this function
should be stored in case the original site and sample data are misplaced. 
Any errors observed should be corrected immediately using the Data Entry
Module.

The output file will always be preceded with a line indicating which data
file is being analyzed, as well as the series within the data file that
have been specified.  The summary information presents: (1) the sample
series number (column number within the data matrix) and the sample ID;
(2) the inner and outer ring dates, as well as the types of ring dates
(pith, innermost, outermost, or bark dates); (3) the length of the sample
(outer minus inner date plus one); (4) the number or rings considered in
any analyses ("number in final analyses" or actual number of recorder
rings for that sample); (5) all information about the fire history for
that sample, including the year, season, and type of fire event (fire
scar or other injury), as well as the fire intervals ("FI") between fire
scars; and (6) all summary information for that particular sample, such
as the total number of fire scars, total number of all indicators,
average number of years per fire, and the sample mean fire interval
(average of all fire intervals for that particular sample).  Note that
fire intervals are given only between years that are designated as events
recorded by fire scars rather than other injuries.  The statistic
"Average number years per fire" is simply the number of valid recorder
years ("Number in final analysis") divided by the total number of fire
scars.  Note that this statistic is different from the "Sample mean fire
interval" which is the average of all fire intervals listed for that
sample.

At the very end of the Summary Information output is the Summary Informa-
tion for the entire site.  This table contains: (1) the beginning and
ending years of the data matrix (and corresponding length); (2) the total
number of samples in the data file; (3) the total number of recorder
years, fire scars, and other indicators for all samples; (4) the average
number of years per fire (total number of recorder years divided by the
total number of fire scars) and per all injuries (total number of
recorder years divided by the total number of fire scars and other in-
juries); (5) the average of all sample mean fire intervals; (6) the total
number of years with fire, along with the corresponding percentage of
years with fire (total fire years divided by total years multiplied by
100) and percentage of years without fire (100 minus the percentage of
years with fire); and (7) the Mean Fire Interval (MFI) based on the
percentage of years with fire (percentage of years with fire divided by
100).  This summary information should not be confused with any
statistical tests performed in other analyses.  This site summary
information is strictly for summary purposes only.

Obtaining Year-by-Year Sample Depth Information
Occasionally, the user would like to be able to graph or analyze the
changing sample depth of the data file over time to help determine when
to begin and end the statistical analyses.  Selecting "S" from the
Summary Menu will create a large table in the output file containing year
by year information about the data file.  The table contains: (1) all
years represented in the data matrix; (2) the number of samples during a
particular year; (3) the total number of actual fire scars, other
indicators, and total number of injuries (fire scars plus other
injuries); and (4) the percent scarred (fire scars divided by number of
samples multiplied by 100).  The year by year listing is useful because
the table can now be imported into any graphics program or spreadsheet
(as an ASCII file, space delimited), and the sample depth can be plotted. 
The user should observe at which point the sample depth becomes "stable,"
or "flat" because it is at this point the user may wish to begin the
statistical analyses.  This is only a suggestion.  The analyses can begin
at any point the user feels sample depth is adequate, and the user may
wish to plot the sample depth alongside the number of fire occurrences in
any 10, 25 or 50 year period (see next section below) to determine not
only when sample depth becomes stable, but also when the number of fire
occurrences first approaches an adequate number.  The listing of fire
scars, other indicators, total injuries, and percent scarred may also
help determine an adequate period for analysis, but is listed for summary
purposes only.

Obtaining Year-by-Year Information on Fire Frequencies
The next three functions in the Summary Menu are used to indicate the
number of fire events that occurred during periods of pre-specified
durations.  To standardize these durations among different studies, the
durations chosen are 10, 25, and 50 years in length.  Selecting "1", "2",
or "3", or any combination of the three, will generate the sample depth
table described in the previous section, with additional columns that
provide information on the number of fire events during the selected
duration.  Note that these are "running" durations in which the periods
always overlap by (n-1) years, where n is the selected duration.  For
example, selecting "2" will generate an additional column (after the
percent scarred column) that shows the number of fire events in "running"
(or overlapping) 25-year periods. The information on fire frequency can
be imported into a spreadsheet or other graphics program, then plotted
year-by-year to show the changing number of fires over time.  Note that
the number of fire events in any 10, 25, or 50-year duration can be
easily converted to fire frequency by dividing the number of fire events
by the duration of the period.  It is important to recognize that no
statistical tests are associated with this summary information.

Obtaining Year-by-Year Information on Percentage Scarred
Selecting "4" through "6" will add columns to the table previously
described that show running calculations for the percentage of trees
scarred.  The percentage of trees scarred can also be calculated for
running 10, 25, and 50 year periods to help determine whether changes
occurred in fire spread patterns.  For example, certain periods may have
experienced more widespread fires in which the majority of trees were
scarred during any one fire year.  Note that these calculations only use
the years in which fire occurred, because including non-fire years in the
calculation of percentage scarred would bias the results.  Therefore,
this analysis is also dependent on the number of fire years that occur in
one of the pre-selected periods.  The user should try difference lengths
for the periods analyzed to determine which of the three are optimal.

Selecting "P" will calculate a "step" function for the percentage of
trees scarred in all fire years, but will override the non-fire years as
if they did not exist.  For example, if the percentage of samples scarred
in 1824 was 55%, this value will be listed for all years until the next
fire year when a new percentage scarred value is calculated.  This
function is useful because effects of "outlier" fire years (e.g., a year
when only 5% of the trees were scarred in a period when the majority of
percentage scarred values was closer to 75%, or vice versa) are
minimized.  Note that this is not a running calculation as the previous
three functions.

INFORMATION ON SEASONALITY

Information on the seasonality of past fires is critical to our
understanding of current (and future) fire regimes.  The intra-annual
position of the fire scar within the annual tree ring provides a clue as
to when during the growing season the fire had occurred.  Determining the
actual season of the fire event will depend on local and regional
conditions, such as the elevation, topography, climatic influences,
forest composition and structure, and past land use history of the site.
For example, the seasonal interpretation for a fire scar in the middle
portion of the earlywood from a sample in the American Southwest may be
different from the interpretation for a fire scar in the same
intra-annual position from a sample in the Pacific Northwest.

While information on the seasonality of fires can be optionally entered
using the Data Entry Module, researchers should obtain this information
from the growth- ring record, and take advantage of the ecological
information offered by these analyses.  During data entry, fire events
may be entered without any seasonal designation by entering "U"
("undetermined") for the season.  In this case, the seasonal analyses
will be superfluous.

The annual ring must be subdivided into cambial growth zones to obtain
seasonal information.  Division of these cambial zones has been 
described previously in the literature, especially in studies conducted
in the Southwestern U.S.  The earlywood is divided into three separate
zones: an early portion (E), middle portion (M), and late portion (L).
Any fire scar occurring in the latewood is designated separately as "A".
In addition, a fire scar can occur between the boundary separating the
latewood of the previous year from the earlywood of the current year.
This represents a fire that occurred sometime after the previous growing
season but prior to the current cambial growing season (during the
period of tree dormancy, designated as "D").

Unfortunately, tree dormancy spans across two years.  Conventions have
been adopted to deal with this problem, and the adoption of additional
conventions may be necessary depending on local and regional
characteristics.  For example, in the American Southwest, fire events
after the cambial growing season are relatively rare in the historical
record because of the onset of the summer monsoon in late summer and a
return to higher moisture levels.  Therefore, any dormant season fire
scar is assigned to the current growing season, or the year containing
the earlywood immediately after the fire scar.  A different situation
exists for the western slopes of the Sierra Nevada where fire history
studies have been conducted for giant sequoia stands.  Historical records
indicate that fires usually occur later in the growing season. 
Therefore, any dormant season fire scar is assumed to have occurred
immediately after the previous growing season, and the fire event is
assigned to the calendar year corresponding to the latewood cells
preceding the fire scar.

If seasonal information has been entered, the user can begin the analyses
by selecting "S" from the Statistics Module Main Menu.  A new window will
open displaying the options within the Seasonal Analysis Menu.  The first
option, "Analyze Seasons," is simply a toggle between "Yes" and "No." 
The next two options allow the user to group together certain seasonal
combinations to provide a broader assessment of the dominant season of
past fire occurrence.  The seasonal analysis can therefore provide a fine
resolution analysis on the individual seasons themselves, as well as a
broader resolution spanning a longer period within the growing season.
The latter is termed the "dominant mode" of seasonal occurrence.

Selecting Season Combinations
By analyzing combinations of seasons, the user can determine whether
fires had occurred predominantly in one longer period or another.  For
example, the user may wish to select those combinations that will provide
information on whether fires occurred predominantly early in the growing
season or later in the growing season.  By default, the Statistics Module
is set for such an analysis, combining the "D" and "E" into an early
season, and the "M", "L", and "A" into a later season.  Of course, these
combinations may differ depending on the site being investigated, and the
user may decide to include the "M" designation in the early season.  In
the giant sequoia habitat type, fires occurred predominantly after the
current growing season and, by convention, these are designated as "D"
fire events assigned to the adjacent latewood cells.  The user
investigating giant sequoia fire ecology may wish to group the "A" and
"D" events into one season and all other events into another to determine
whether a period existed in the past when fires occurred earlier than the
end of the growing season.  Any temporal and spatial change in the
seasonal occurrence of fires may be attributable to changing patterns of
climate, vegetation, or human sources of ignition.

The user may analyze two sets of combinations.  To enter combinations for
either set, select either "1" or "2" from the Seasonal Analysis Menu.  A
new window containing an expanded dialog box will open in which the user
should enter the letters for the seasonal combinations.  The user can
enter any combination of the five valid seasons: D, E, M, L, or A.  Note
that any other letters that are not one of the five valid letters are
ignored by the software.  Also note that any "doubles" are ignored as
well (e.g. entering "DEMD").  Letters may be entered in either upper or
lower case.  Once the combination has been entered, the user should
repeat the process for the other combination.  The seasonal designations
for the combinations are then displayed in the Seasonal Analysis Menu.
At this point, the user may select "G" to begin the seasonal analyses.

Intra-annual Distribution and Temporal Pattern
The output from this analysis is divided into three sections.  The first
provides information on the intra-annual distribution and temporal
pattern of fire events during any one year.  All fire years are listed,
one per row, over the range specified by the user, along with the seasons
determined from the fire scars arranged in temporal order beginning with
the earliest designation ("D") and ending with the undetermined
designation.  In addition, total sample depth is also given during each
fire year, completed with the right bracket character.  Hyphens complete
the information on each fire year by indicating the entire number of
samples in the matrix.  Note that the sample depth given can also aid in
the determination of an adequate period to analyze, in conjunction with
the Sample Depth Table produced in the Summary Information.  Sample depth
information in the seasonal summary is not as comprehensive as the Sample
Depth Table, but is easier to interpret.

This intra-annual information can provide clues to the behavior of fire
during any given fire year.  For example, a wide range of seasons for
any given fire year may indicate that the fire began early (perhaps
indicated by some dormant season designations), and lasted late into
the growing season (as indicated by some late season designations).
Fire during this year would contrast with fire during a year when nearly
all fires scars indicated fire occurred during only one portion of the
season.  This intra-annual distribution can be plotted as bar graphs with
other graphics software or spreadsheets to visually assess possible fire
behavior during the fire year to help the user analyze the probable range
of months the fire burned in a particular area.

Dominant Mode of Seasonal Occurrence
The second section provides additional information on the intra-annual
distribution and the dominant mode of seasonal occurrence.  Each fire
year within the specified range is once again listed, one per row, along
with the percentage of fires that occurred within the individual seasons. 
As before, information in this section is only given provided seasons
were entered - no information will appear if seasons could not be
determined.  The percentages allow bar graphs to be created (with other
graphics packages) similar to those available from the information
provided in the first section without using absolute numbers.  The
dominant mode of seasonal occurrence is provided in the bar graphs to the
right side of this section using the group combinations selected by the
user.  The user should carefully inspect this section to determine
whether changes in the seasonality of fire have occurred over time.  Any
change in the dominant mode of seasonality could indicate the influence
of changing climate on local fire regimes.

Summary Information on Seasonal Analysis
The last section provides a summary of the seasonal information contained
in the data matrix over the specified range.  The total number and
percentage is given for the number of fire scars in which seasons could
be determined relative to the total number of fire scars, as well as the
number and percentage of fire scars in which seasons could not be
determined.  As with pollen analysis, the number of fire scars in which
seasons can not be determined may provide additional information on the
nature of preservation of fire-scarred material at that particular site,
something researchers should consider in future field work in the area
(i.e. take additional care during the cutting and sampling).  The same
information is provided for each individual seasonal designation, and,
finally, for the group combinations selected by the user.

FIRE INTERVAL ANALYSES

The analysis of fire interval information is perhaps the most important
for analyzing a fire regime, because it provides information on how often
fires occurred in the past prior to and during periods of human
disturbance.  By evaluating the intervals between fires that occurred in
the past, researchers can help land management agencies develop better
management plans for dealing with the probable occurrence of fires in the
future.  In addition, the analysis of fire intervals can provide
information on the degree of fire hazard that exists for a certain area
given the time since the last major fire.

One advantage of the FHX2 system is the use of the Weibull distribution
to model fire interval data.  The Weibull distribution was developed to
provide better models for data measuring time to mechanical failure.
Engineers found the mean time between failure for mechanical devices did
not follow a normal distribution, and that analyses of failure informa-
tion based on the normal distribution would be biased and incorrect.
For example, if the data were not symmetrically distributed, then the
mean of the distribution would not be an adequate measure of central
tendency in the data.  The Weibull distribution was named after the
engineer who first described and used a new distribution that was able
to better model data that was either highly negatively or positively
skewed, yet was flexible enough to also model normally distributed data
and negative exponential distributions.  The power of the Weibull
distribution therefore lies in its flexibility.  It can fit a variety
of shapes to data over a wide range of distributions.  Fire ecologists
have long suggested more flexible distributions be used to model fire
interval data other than the normal distribution.

Choosing Percentage-Scarred Classes
To begin the fire interval analyses, select "F" from the Statistics
Module Main Menu.  A new window containing the Fire Interval Analyses
Menu will open.  Analyses can be conducted over a wide range of
percentage-scarred classes, including analyzing all fire years,
regardless of the percentage scarred, to only those fire years in which
nearly all samples collected were scarred.  The percentage-scarred
classes listed designate the most frequently used percentage classes, and
should be considered standard when conducting fire interval analyses.  To
select a particular percentage scarred class, simply type the number
listed to the left of the class.  The toggle to the right will switch to
"Yes."

Various percentage-scarred classes can be analyzed.  All classes may be
selected by selecting "Y" from this menu, while all classes can be
de-selected by selecting "N."  Occasionally, the user may wish to include
the last incomplete interval at the end of the temporal sequence, because
this interval may contain additional information on the fire regime.  For
example, in fire regimes that have been severely disturbed by humans, the
last fire scar in the data matrix may have occurred in the late 1800's
even though numerous samples from living trees were collected to help
evaluate 20th century fires.  The long fire-free period between the last
fire in the late 1800's and the year in which the samples were collected
is a fire interval the user may wish to consider including in the
analyses.  To include the last incomplete fire interval, the user can
simply select "I" from the Fire Interval Analysis Menu.  The toggle to
the right will switch to "Yes," and the inclusion of this incomplete
interval will be mentioned in the output.

Including Other Injuries
Choosing this option will allow other injuries to be treated as fire
scars because, in many instances, other injuries (e.g. excess resin
production in a tree ring, or expanded latewood in giant sequoia tree
rings) can be considered as valid as fire scars for indicating fire
events.  To include other injuries in the analyses of fire intervals,
select "O" from the Fire Interval Analysis Menu.  The toggle will switch
to "Yes."  When the analyses are activated, other injuries in the data
matrix (denoted by lower case letters) will be treated as if they were
fire scars, and will change the intervals included in the analyses
because the percentage of trees "scarred" will be changed as well.

Threshold Level
Choosing this option allows you to set the level that FHX2 uses to
identify significantly short and long intervals.  While it is tempting to
simply use the .025 or the .05 levels, it should be recognized that these
levels may be too stringent for ecological interpretations.  The FHX2
system uses the .125 level by default, a convenient level because it
delimits the 25% of all intervals (12.5% on each tail) that are
considered outside the normal range of variability.

Minimum Samples Scarred
The next option also affects the number of intervals included in the
analyses.  Often, the user will wish to include only those intervals that
result from major fire events as denoted by the absolute number of trees
scarred as opposed to the percentage.  For example, the user may wish to
include only those years in which at least five samples were scarred
during the specified range of years.  This option is simply one of the
filters as described in the Composite Axis Menu in the Graphics Module. 
The minimum number of samples scarred may be changed by selecting "M"
from the menu.  A dialog box will then open and the user can enter the
new minimum number of trees scarred to include in the analyses.  In the
resulting analyses, only those years will be included in the analyses
that satisfy the minimum requirement that has been set by the user.  If
the user wishes to include all fire events, regardless of the number of
trees actually scarred, the user should first choose to include the other
injuries, then set the minimum requirement to zero.

Number of Intervals to be Graphed
The user also has control over the number of intervals to be plotted in
both the frequency distribution and cumulative frequency distribution
diagrams.  By default, this value is set to 30, so that the number of
intervals in each interval class will be plotted up to a maximum that
corresponds to a 30-year interval.  The number of intervals greater than
30 years will be grouped together in the last class interval.  To change
the maximum, the user can select "M" from the Fire Interval Menu, and
enter a new maximum.  The largest interval that can be graphed is 99
years.  Setting the maximum value to be graphed to the maximum interval
encountered (as observed in the descriptive statistics discussed below)
allows a finer resolution to be displayed and imported into a graphics
programs.

Analyzing Site Composite or Sample Information
The user may examine the fire intervals of the data set by two methods:
(1) fire intervals derived from the site composite (the default); and (2)
fire intervals derived from the individual samples (point intervals).  By
far, the most common approach is to analyze the composite information on
fire intervals as derived from the site composite.  However,
occasionally, the user may wish to analyze "point intervals" as opposed
to "site intervals."  To conduct this type of analysis, select "C" to
toggle between analyzing the site composite or point samples.  Note that
this analysis can only be conducted on the "All Scarred" class because
fire intervals will be derived from individual samples as opposed to the
composite of all samples.  Once chosen, all other options are
deactivated.  Foe example, if the user had selected to analyze the All,
10, and 20 percent scarred classes, then chose to analyze individual
samples, the options for the 10 and 20 percent scarred classes would
toggle to "No."  Comparing the results from both the site and point
interval analyses may provide information on the areal extent of fires
and the ability of fire to spread within the site.  If results indicate
that the point intervals are much longer than the site intervals, then
some inference can be made on the ability of fire to spread within the
site.

Creating a Spreadsheet Table
Another option in the Fire Interval Menu allows the user to create a
spreadsheet table containing all basic statistics generated by the
statistical analyses.  To activate this option, select "T" from the menu,
and the toggle will switch to "Yes."  When the analyses are run, the
program will create a file named "fhstats.tab" that contains: (1) the
site name; (2) the beginning and ending years of the site, as well as any
subperiod specified by the user; (3) the number of series analyzed; (4)
the  percentage scarred class; (5) the total number of intervals; (6) the
mean, median, minimum, and maximum interval; (7) the standard deviation,
coefficient of variation, skewness, and kurtosis of the data set; (8) the
Weibull shape and scale parameters; (9) the Median Probability Interval,
and its associated frequency probability; (10) the Maximum Hazard
Interval; (11) the Maximum Density Interval; (12) the Lower and Upper
Exceedance Intervals (LEI and UEI).  This information is presented in a
tabular format.

Note that if the option to create the table is activated, and the file
already exists from a previous run, the information will be appended to
the end of the file.  This allows the user to create a table, ready for
importing into a spreadsheet or word processor, that contains the summary
information from numerous sites, as well as for different percent scarred
classes, or over different time intervals, or based upon a different
subset of samples within the main data set.

Creating a Data File for Plotting
By selecting "P" from the Interval Analysis menu, the user can choose to
create a file containing all the data from the fire interval analyses
used to plot the curves for various functions.  This feature is
convenient because the file can then be imported into a spreadsheet or
other graphics program for plotting various curves.  The following infor-
mation is output to a file named "plot.tab":

   Column 1: Contains the fire intervals from zero to the maximum
             interval.  Label = "int".
   Column 2: Contains the actual number of fire intervals in that parti-
             cular interval class.  Label = "total".
   Column 3: Contains the relative frequency of the actual intervals in
             Column 2, such that the total in this column equals 100.0.
             Label = "rel".
   Column 4: Contains the cumulative frequency function for the actual
             fire interval data.  Label = "C(t)".
   Column 5: Contains the probability density function of the Weibull
             distribution fit to the actual data.  Label = "f(t)".
   Column 6: Contains the cumulative frequency distribution of the
             Weibull distribution fit to the actual interval data.
             Label = "F(t)".
   Column 7: Contains the complement of F(t), or the mortality function
             of the Weibull distribution.  Label = "A(t)".
   Column 8: Contains the hazard function of the Weibull distribution.
             Label "h(t)".
   Column 9: Contains the K-S statistic, so the user can visually
             identify the area which has the greatest difference between
             the two cumulative functions.  Label = "K-S d".

The file is comma-delimited, and most spreadsheets should be able to
directly import this file.  The labels are also denoted as such using
quotation marks.  Note that various functions can be plotted on the
same y-axis, such as C(t), F(t), A(t), and h(t), because these functions
range from zero to one (although the hazard rate can exceed one).

OUTPUT PRODUCED BY THE FIRE INTERVAL ANALYSES

Once "G" has been selected, the results from the analyses are written to
an output file, read by the FHX2 software, and displayed on screen.  The
results from these analyses are comprehensive, consisting of eight
sections in two major parts.

Fire Intervals in the Analysis
The first section of the output lists the fire intervals used in the
subsequent analyses.  The section lists each fire year that meets the
requirements (i.e. filters) set by the user regarding the percentage
scarred, other injuries, and minimum number of samples scarred.  To the
right of each year, the total number of trees scarred, the total number
of recorder trees, the percentage scarred, and the fire interval are
listed in separate columns.  Remember that it is the last column of
information, the fire intervals, that will be analyzed.  If the user
selected to include the last incomplete interval, this will be indicated
at the very bottom of the table.  Also note that the number of fire
intervals used in the analyses is always one less (n - 1) that the number
of fire years that meet the requirements set because no fire interval is
associated with the first fire year (i.e. no prior fire year is available
to allow an interval to be established).

Frequency Histogram of the Actual Data
The next section provides a frequency histogram of the actual fire
interval data.  A frequency histogram graphically displays the shape of
the actual distribution as a series of bar graphs (here, horizontally). 
Each bar represents the number of fire intervals of that particular
length in years.  The very left hand column of this section lists the
fire interval itself.  The next column lists the actual number of fire
intervals of this particular length.  The third column lists the percent-
age of fire intervals of this particular length.  By default, the
frequency distribution table will list all intervals up to length 30
years, after which it will simply list the number of all intervals
greater than 30 years in length in one line to conserve space in the
output file.  The range of intervals displayed can be set by the user in
the Fire Interval Analysis Menu.

This portion of the output should be visually inspected to assess how
much the distribution deviates from a symmetric distribution. In a
distribution that is not skewed, the mean and median would coincide.

Kolmogorov-Smirnov Test for Goodness-of-Fit
Prior to interpreting any results from the fire interval analyses, the
user must first investigate the distribution of the fire interval data to
determine whether this distribution can be modeled with other theoretical
distributions (e.g., the normal distribution).  Several parametric and
non-parametric statistical tests have been devised to help determine
whether a distribution approximates other distributions.  One of the most
frequently used is the Kolmogorov-Smirnov (K-S) test.  Because the shape
of theoretical distributions is known, it should be possible to compare
the shape of the distribution of the fire interval data to hypothetical
distributions.  The K-S test uses the two cumulative frequency
distributions (one from the actual fire interval data and the other from
the hypothetical empirical distribution) to perform the test by computing
the d-statistic, the maximum difference between the two distributions in
any particular interval.  This analysis uses the empirical distribution
because this is a step function that is often used to approximate the
sigmoidal shape of the highly complex cumulative distribution function
for the normal distribution.  However, normality, or lack therefore,
should be assessed with caution using this analysis.

Because the actual data are being compared to a hypothetical
distribution, it is also possible to determine the probability of
obtaining a higher d-statistic simply by chance.  As with any other
statistic, a correction must also be made for the number of observations
used in calculating the statistic by adjusting for loss of degrees of
freedom.  The more observations used to test the statistic (i.e. the more
fire intervals obtained for the analysis), the more robust will be the
statistic.  The user should keep in mind that, if two distributions are
very similar, then the d-statistic will be very small.  If the number of
observations used are high (e.g. greater than 20), then the probability
of obtaining a d-statistic equal to or greater than the observed value
will also be very high.

The first three lines are known as decision rules, commonly used in
statistical analyses to help interpret results.  The first line states
the null hypothesis, designated as "Ho."  The null hypothesis is the
hypothesis of no difference between the distribution of the fire interval
data and the empirical distribution.  This hypothesis will be tested
using the K-S d-statistic.  If the statistic is significant, indicating
that obtaining another statistic higher than this simply by chance is not
very likely, then the null hypothesis is rejected and the alternate
hypothesis (designated as "Ha") is accepted (the data do not represent an
empirical distribution).  The user must keep in mind that similar distri-
butions will have a low d-statistic and, subsequently, a high probability
of obtaining a value greater than this simply by chance.  In general, the
user will often find high values for the d-statistic when investigating
fire interval data, indicating the data are not normally distributed.

Two or Three Parameter Weibull Distribution Parameters
The next step in the analyses of fire interval data is to fit a Weibull
distribution to the data.  As with all distributions, the Weibull
distribution is a combination of three different parameters: the shape
parameter (c); the scale parameter (b); and, the location parameter (a). 
This class of distributions is known as three-parameter distributions. 
Often, distributions can be simplified, or reduced to two-parameter
distributions, by setting one of the three parameters to zero.  This is
the case with fire interval data.  Because no fire interval can be less
than zero, the location parameter (where on the x-axis of the coordinate
system the distribution is located) may not be necessary, which greatly
simplifies the calculation of the other two parameters.  However, the
user has the choice to model either a two or three parameter
distribution.

The scale parameter will indicate the range of values along the x-axis
encompassed by the Weibull distribution, and is therefore measured in
years.  Approximately two-thirds of all data contained in the interval
analysis will be less than the value of the scale parameter.  This
property is useful because a quick look at the scale parameter provides
a crude measure of the range of years being modeled by the Weibull
distribution.  The shape parameter is unitless, and provides information
on the overall form of the distribution being fitted to the data.  A
value of 1.0 indicates that the data conform to a negative exponential
distribution, while a value near 3.4 indicates the data approximate a
normal distribution.  The latter case may be verified by looking at the
results of the previous K-S test.  In general, values for the shape
parameter will often range between 1.5 and 3.5.

Equations and algorithms used to calculate all parameters of
distributions can be extremely complex.  Methods used to calculate the
two parameters for the Weibull distribution in the Statistics Module were
obtained from manuals on hydrology, where the fitting of distributions to
data is extremely important.  The first line provides the cumulative
distribution function of the Weibull distribution, showing the
relationship among the three parameters.  The shape and scale parameters
are then listed.

Kolmogorov-Smirnov Test for Goodness-of-Fit: Weibull Distribution
The next stage of the analysis is to determine how well the Weibull
distribution, with the derived shape and scale parameters, fits the fire
interval data.  For this, we turn once again to the K-S test because we
wish to compare two distributions: the cumulative frequency distribution
of the original data set, and the cumulative frequency distribution of
the derived Weibull distribution.  The test statistic is calculated in
the exact same way as in the previous example when determining whether
the data approximate the empirical distribution.  The only difference is
the new set of decision rules.  In the case of the Weibull function, a
low value for the d-statistic indicates a close match between the two
distributions, the desired outcome.  In this case, the probability of
obtaining a higher d-statistic would be very high, causing us to accept
the null hypothesis of no difference between the original data and the
Weibull distribution.

Weibull Distribution Exceedance Probability Table
Once the Weibull function has been fit to the fire interval data, and the
Weibull distribution has been deemed a superior model as opposed to the
normal distribution, then important statistical descriptors can be
derived.  In the Exceedance Probability Table, various probability levels
are listed in the left hand column, while the hypothetical fire interval
associated with that probability level is listed in the second column.  
The most important property of this table is the fire interval associated
with the 50% level of exceedance probability.  This interval is termed
the Median Probability Interval (MPI) and provides a least biased measure
of central tendency in skewed distributions of fire interval data.  The
MPI is analogous to the MFI based on the normal distribution.

The Exceedance Probability Table provides a listing of statistically
significant small and large fire intervals.  In the FHX2 system, the .125
level of significance is used, but any level can be established by the
user.  Any fire interval associated with the 87.5% exceedance probability
level or higher will therefore be a significantly small fire interval
(one-tailed test). Note that the significantly small intervals are
denoted by the "less than" sign ("<") while the significantly large
intervals are denoted by a "greater than" sign (">"). 

Two-Parameter Weibull Distribution Probabilities
The final stage of the fire interval analyses determines the
probabilities associated with the actual fire interval data obtained from
the data set.  In the first column, years are listed in which fire
occurred according to the filters set by the user.  Fire intervals are
listed in the second column.  It is this column on which the analyses
have just been conducted.  In the third column, the probabilities are
listed for each fire interval as determined by the Weibull function. 
Note that significantly small and large intervals are denoted by "less
than" and "greater than" signs in the second column.

The next two columns provide additional information obtained about the
distribution of the fire interval data based on the Weibull function. 
These are the hazard function, h(t), and the probability density
function, p.d.f.  All distributions can be thought as consisting of four
separate functions: the cumulative distribution function (c.d.f.), which
we have already discussed; the survivorship, or survival, function, equal
to 1-c.d.f.; the probability distribution function (p.d.f.), in which all
values underneath the curve of the distribution equal to one; and the
hazard function, which consists of one of two expressions in the c.d.f. 
The hazard function is perhaps the most useful from the standpoint of
fire interval data, because it provides a measure of the hazard
associated with a fire interval of given length.  The hazard function
increases with longer fire intervals.  Note that the hazard function is a
separate function apart from the significantly small and large intervals
determined using the c.d.f. in the previous table, and should be used
independently from any other statistics.  The p.d.f. is supplied to
provide additional diagnostic information, and is used to isolate the
Maximum Density Interval.

The hazard function is sensitive to extremely large fire intervals, and
the user should carefully select the range of years considered
representative of the fire regime prior to any human disturbances.  A
more realistic measure of the hazard function would be provided by
analyses conducted on a period ending in 1880.  A useful measure is the
maximum length of the fire interval in which the habitat can go without
burning.  To compute this interval, a value equal to (1-threshold) is
substituted into the equation for deriving the hazard function, and the
fire interval associated with this value is determined.  The result is
termed the Maximum Hazard Interval.

Summary Descriptive Statistics for Actual Data
The next section in the output file lists the simple descriptive
statistics for the actual fire interval data.  These descriptors include:
1) the total number of fire intervals used in the subsequent analyses;
2) the Mean Fire Interval;
3) the median value of the fire intervals, representing the mid-point
value in the fire interval data array (often a better indicator of
central tendency in the distribution than the mean, especially for
distributions that are not normally distributed);
4) the Maximum Density Interval, or the fire interval that provides the
greatest amount of area under the pdf, and is therefore equivalent toa
modal measure;
5) the Median Probability Interval, the measure of central tendency based
on the Weibull distribution;
6) the fire frequency of the MPI, or 1/MPI;
7) the standard deviation of the fire interval data (the square root of
the variance - 68% of all data points will fall within one standard
deviation above and below the mean value, while 95% will fall within plus
and minus two standard deviations);
8) the coefficient of variation (cv) of the fire interval data, simply
the mean divided by the standard deviation.  The cv is used to compare
distributions that have different means and/or variances, thus providing
a descriptor  not affected by the scale of the data.
9) the skewness of the fire interval data.  This descriptor is useful for
determining the degree to which a distribution is not normal.  A
distribution that has larger data values to the right of the mean will
cause the distribution to be positively skewed (often the case with fire
interval data as the distribution is truncated to the left of the
distribution because no value can be less than one;
10) the kurtosis of the fire interval distribution, a statistical
descriptor that indicates the amount of "peakedness" in the distribution. 
Data that are clustered about the mean with a few large and small
intervals create a distribution that is "peaked."  Values less than zero
indicate a distribution that is platykurtic, or flat, while values
greater than zero indicate a distribution that is leptokurtic, or peaked;
11) the Weibull location parameter, if a three parameter model was
specified;
12) the Weibull scale parameter;
13) the Weibull shape parameter;
14) the minimum fire interval;
15) the maximum fire interval.  These last two descriptors can be used to
measure the range of the actual fire interval data.
16) the Lower Exceedance Interval, or the interval that delimits a
significantly short interval based on the Weibull fit;
17) the Upper Exceedance Interval, or the interval that delimits a
significantly long interval based on the Weibull fit; and,
18) the Maximum Hazard Interval.

TESTS FOR TEMPORAL CHANGES OF THE FIRE REGIME

An important analysis of fire regimes is to determine whether a change
has occurred in some aspect of the fire regime at some point in time.
The first clue may perhaps be seen visually when examining the master
fire chart created for a site.  For example, a researcher may see a
change in the length of the fire-free periods between the early part of
the record and a later part.  Or, the researcher may notice that the
percentage of trees that had been scarred appears to have changed between
one period and another.  The central question is whether these changes
are statistically significant.  The researcher should not, however, rely
solely on the statistical significance of these tests.  Changes may be
apparent in the fire regime, but the changes may not be statistically
significant.  The tests for statistical significance only add a measure
of robustness when evaluating the degree to which changes have occurred.

Two aspects of the fire regime can be tested between two different,
non-overlapping periods.  First, the researcher may test whether or not
changes have occurred in the fire intervals between fires.  For example,
an earlier period may appear to have a longer average between fire years
than does a later period.  If changes in the average length of fire-free
periods have occurred, then the researcher should investigate possible
reasons why these changes have occurred.  Possible explanations could
include changes in sample depth, subtle changes from one climate regime
to another, changes in land-use or land management practices, or changes
in the composition of vegetation and the fuel complex.  Second, the re-
searcher may test whether the percentage of trees being scarred by fires
has changed between two periods.  If changes in the number of trees being
affected by fire have occurred, this will provide information on the
nature of the fuels, the capability of fire to spread, and/or the
maturity of the forest being affected.

Three aspects of the temporal and spatial characteristics of the fire
regime can be tested: (1) changes in the mean fire interval, or the mean
percentage of trees being scarred, between non-overlapping periods; (2)
changes in the variability about the mean; and, (3) changes in the
overall distribution.  Three basic tests are used in these analyses, as
well as in the analyses for differences in spatial characteristics of the
fire regime (discussed in the following section).  To test for changes in
the mean interval, or the mean number of trees being scarred, from one
period to the next, the FHX2 system uses the Student's t-test.  To test
for changes in the variability about the mean interval, or about the mean
number of trees being scarred, from one period to the next, the system
uses the folded-F test.  Finally, to test for changes in the overall
distribution of the fire intervals or the percentage-scarred between
two periods, the FHX2 system uses the Kolmogorov-Smirnov test, already
introduced in the previous section.

Results from t-tests and folded-F tests are unbiased only if the data are
normally distributed.  Fire interval data are rarely normally
distributed.  The same property holds for percentage-scarred data.
Because the percentage of trees being scarred has a lower bound (zero)
and no upper limit, the data tend to be positively skewed.  Because the
tests assume normality in the data, we must somehow transform the
original data into a new data set that approximates normality.  This
transformation is accomplished using moments of the distribution itself.
The algorithm used for the transformation is based on the third and
fourth moments of the original data distribution (i.e. the skewness and
kurtosis of the distribution which indicate the departure of normality).
No one-step transformation ever completely transforms a data set to one
whose distribution contains zero skewness and kurtosis.  However, it is
not the transformation to perfect normality that matters, but rather the
transformation of a distribution to one that approximates normality.
Sometimes, the transformation has little effect.

One additional transformation takes place for ease in interpretation of
the final results.  After the original data (i.e. the fire interval data
or the percentage-scarred data) have been transformed to normality, the
data are then transformed into the standard normal distribution by simply
subtracting the mean from each transformed data value and then dividing
by the standard deviation of the distribution.  This transformation gives
the data set a mean of zero and standard deviation of one, useful for
comparing fire regimes from different sites.  The additional transforma-
tion can be viewed as simply an additional term in the equation for the
first transformation.  The transformations should be of little concern to
the user, because the transformations are simply the application of an
equation to each data value, much like adding a constant value of one to
each data value to increase the overall mean.  The final outcome of the
transformation process is a data set that approximates the standard
normal distribution on which the tests may now be applied.

Beginning the Tests for Temporal Changes
To begin the tests for changes in temporal aspects of the fire regime,
select "T" from the Statistics Module Main Menu.  The Temporal Changes
Menu will then open displaying the options available to the user.  The
first options denote what is to be tested, either changes in the fire
intervals or changes in the percentage of trees scarred.  To choose fire
intervals, select "F" from the menu.  Likewise, to run tests on the
percentage of trees being scarred, select "P."  Toggles to the right of
each selection will indicate the user's choice.  Note that tests on both
aspects of the fire regime, fire intervals and percentage scarred, can be
tested during one run.

Selecting the Periods to be Tested
The next two options allow the user to select which periods are to be
tested.  The periods are divided into a "control" period and a "test"
period.  The control and test periods are not meant to imply that one
period has been experimentally controlled while some changes were
artificially introduced into another period.  The terms are simply used
to denote the periods, and could easily be considered as "Period One" and
"Period Two."  However, the earlier period is here denoted as the
"control" period because we are assuming that a change has taken place
during the second period, which is to be statistically tested versus the
first period.  By default, the Statistics Module evenly divides the range
of years specified by the user into these two periods.  The user,
however, has complete control over the range of years in each of the
control and test periods.  The user should carefully inspect the graphs
of the fire history for the site to choose which periods are to be
tested, and should also carefully consider the possible effects of small
sample sizes.  The tests are more robust for large sample sizes,
preferably above 20 data observations in each period.

To change the beginning or ending years for each period, the user can
select "C" for the control period or "T" for the test period.  A menu
will open, and the user should select whether to change the first or last
year.  A dialog box will then open and the user should type in the
desired year.  Once the years have been input, the changes will be seen
in the Temporal Changes Menu.  The default division between the control
and test periods can be returned by selecting "R" from the Temporal
Changes Menu.  One important note must be emphasized.  It is critically
important that the periods be independent from each other.  The periods
must not overlap as the two data sets must be mutually exclusive for the
statistical tests to be valid.

Selecting Minimum Percentage-Scarred Classes
When analyzing both the probable changes in fire intervals and the number
of trees being scarred, the user has control over the fire years which
enter into the analyses and, therefore, the fire intervals between the
years and the percentage-scarred classes as well.  This option is made
available by allowing the user to specify a minimum percentage scarred
class to designate only certain years that become included in the
analysis.  This option is similar to the option for the Composite Axis in
the Graphics Module where the user had the option to plot only those
years on the Composite Axis in which the percentage scarred was equal to
or greater than a certain user-specified level.  This allows only certain
years to be graphed on the Composite Axis.  For example, in the tests for
temporal changes, the user may wish to analyze only those years in which
fire scarred at least 10% of all trees sampled.  By using this
percentage, the smaller fires which scarred only a few trees in any given
fire year will be excluded.  Similarly, the user may wish to analyze only
those major fire years in which 50% of all trees were scarred by fire.

To specify a minimum percentage scarred class, the user can select "I" to
change the minimum percentage scarred class for fire interval analysis,
or "S" to change the minimum percentage scarred class for analyzing the
number of trees being scarred.  A dialog box will open and the user
should input a new percentage.

Changing the Minimum Number of Samples
Another filter the user may wish to consider allows only certain fire
years to enter the analysis if a minimum number of trees exists for that
fire year.  This option is similar to the option contained in the
Graphics Module for the Composite Axis, and in the Fire Interval Menu
discussed above.  Only those fire years will enter into the analysis if
the minimum number of trees scarred by fire has been met, which
subsequently will have a direct effect on the data arrays that contain
either the intervals between fires, or the number of trees being scarred
by fire.  Note that by designating both the minimum percentage scarred
class and a minimum number of trees scarred by fire, the user has a large
degree of control over the years to be included in the analyses.

Specifying Whether to Include Other Injuries
As seen in the Graphics Module, the user can decide whether to include
other injuries in the analysis and have them treated as if they were fire
scars.  Normally, no statistical analyses are conducted on the other
injuries (designated with lower-case letters in the data matrix) as fire
scars are considered the only direct evidence of a fire event.  In
certain situations, the other injuries may be as diagnostic of a fire
event as a fire scar, as discussed previously concerning the fire history
of giant sequoias.  In this case, the researcher should consider
including other injuries in the analyses.  To accomplish this, the user
can select "O" from the Temporal Changes Menu.  The toggle to the right
will reflect the change.  Choosing to include the other injuries in the
analyses may dramatically change the results, because different fire
years are included that will have a direct effect on the values contained
in the fire interval and percentage scarred data arrays.  When choosing
to include other injuries, the FHX2 software first determines whether the
minimum number of trees scarred has been met.  If so, the other injury
events are converted to fire scar events, and the percentage scarred is
then computed to determine whether it meets the minimum percentage
specified by the user.  Only after these conditions are met is the fire
year included in the analysis.

Interpreting the Output
The first part provides a statistical description of either the fire
intervals or percentage scarred information for both the control and test
periods.  Statistics include: (1) the range; (2) the number of
observations (Num Obs); (3) the mean of the period; (4) the standard
error of the mean (Std Err); (5) the variance about the mean; (6) the
standard deviation (Std Dev); and, (7) the coefficient of variation (cv). 
After the data from the two periods are pooled and transformed, the
results of the transformation described in the previous section are
presented.  Statistics mentioned include: (1) the mean; (2) the standard
error of the mean; (3) the variance; (4) the standard deviation; (5) the
skewness; and (6) the kurtosis.  The latter two statistics should be
examined as these give some indication as to the degree the distribution
departs from normality.  The third portion of the output shows the
results of the transformation once the data have once again been
separated back into a control and test period.

After the information about the transformation process has been printed,
the results from the statistical tests are then displayed in the next
three parts, one each for the three statistical tests described earlier. 
Note that decision rules are printed for all three tests to help the user
interpret the results.  The decision rules are similar to those already
described in the section on fire intervals, and involve the set up of a
null hypothesis (i.e. "Ho," no difference between the two periods), and
an alternate hypothesis (i.e. "Ha," a difference does appear to be
present).

The first section provides information about the results of the t-test
for statistically different means between the two periods.  Note that two
t- statistics are printed.  The first t-statistic should be used under
the assumption of equal variances, as noted in the decision rules.  The
second should be used under the assumption of unequal variances.  To
determine which to use, the user should examine the results of the
folded-F test to determine whether the variances between the two periods
are equal or unequal.

The second part of the statistical tests involves the possible changes in
the variance of data values about the mean.  This test measures the
dispersion of data values about the mean, as the range of values may have
changed between the two periods.  The test statistic is the folded-F
value where the larger of the two variances is placed over the smaller
and the ratio computed.  A very large value in the numerator respective
to the denominator will result in a large ratio and possibly a statisti-
cally significant result.

The final test is the K-S test which uses the d-statistic.  The K-S test
simply determines the maximum distance between any interval year between
the two cumulative frequency distributions of the two periods.  A large
difference would indicate that the two distributions are very different,
further indicating that a change in the fire regime between the two
periods may have occurred.  The user may occasionally find that the K-S
test indicates a statistically significant difference between the two
periods when the previous two tests did not.  The opposite may also
happen.  The t-test and folded-F value may be significant while the K-S
test is not.  It should be remembered that the K-S test is non-parametric
and not as robust as the two parametric tests.  However, any significant
result from the K-S test should be viewed as indicating some difference
between the two periods.

TESTS FOR SPATIAL DIFFERENCES BETWEEN FIRE REGIMES

The first clue that spatial changes may have occurred in the fire regime
may be observed in the master fire chart for the site.  The user should
carefully examine the graph to determine whether a group of samples
differs in some aspect from the remainder of the samples.  For example,
the first 15 samples collected at a site may represent samples collected
higher in elevation than the remaining 20 samples.  The investigator may
wish to determine whether fire regimes of the two areas markedly differ. 
Alternatively, some samples may have been collected in a habitat type
that has a lower fuel loading than the type from which the remaining 20
samples were collected.  The investigator may wish to determine whether
the different habitat types have an effect on fuel loadings and the
resulting fire regime.

Two sites may be combined into one data file using the Merge function in
the Data Entry Module to determine whether the fire regime from one site
differs markedly from another site.  In reality, the composite
information from a number of sites can be combined into one data file,
allowing the researcher to analyze multiple sites versus multiple sites. 
It is becoming more and more common to collect between five to ten
individual sites from a region to obtain information on the spatial
characteristics of the fire regime, and the FHX2 system can easily
accommodate any number of sites using merged Composite Axis information
from these many sites.

It may be necessary to manually group the series differently if the
grouping as originally entered is not conducive to spatial analyses.  One
group must occupy the top part of the matrix, while the second group must
occupy the bottom part of the matrix.  If the samples are interspersed
among each other, the user should use the Data Entry Module to move the
series to the desired locations within the data matrix, making sure that
one specific group occupies the top part and the other group the bottom
part.

Beginning the Tests for Spatial Differences
To begin the spatial analyses, select "A" from the Statistics Module Main
Menu.  The options available from this new menu are almost exactly as
seen in the previous menu for the Temporal Changes Analyses.  The spatial
analyses do not use a control or test period, as the analyses are
conducted over the entire period specified in the range of years by the
user.  In addition, the user must specify after which series to divide
the site.  For spatial analysis, there must be two groups of samples, and
the groups are divided at the dividing series which the user must
specify.  By default, the software chooses the middle series for the
division, but this is rarely the correct divisor series.  To help the
user decide on which series to divide the grouping, select "S" from the
Spatial Differences Menu and all series will be displayed.  The user
should enter the series number (i.e. the column held in the data matrix)
that appears to the left of the sample ID.

Interpreting the Output
The tests used are the same tests as used in the Temporal Changes
Analyses: (1) the Student's t-test for analyzing differences in means;
(2) the folded-F test for analyzing differences in variances; and (3) the
K-S test for analyzing differences in the two distributions.  The output
is interpreted in exactly the same manner as the output for the Temporal
Changes Analyses, and the user can simply refer to the previous section
for that discussion.  Note, also, that two aspects of the fire regime can
be tested spatially as well as temporally: changes in the intervals
between fire, and changes in the percentage of trees that were scarred by
fire.  The investigator should have some a priori knowledge on possible
reasons why differences exist before beginning these analyses.  If
differences appear to exist visually, the user should evaluate possible
differences in topography, such as southern versus northern aspects, and
past and present land management practices, including degrees of past
grazing.  Field notes should be detailed enough to help determine the
differences.

Synchroneity of Fire Dates Between or Among Sites
The last suite of tests concerns the synchroneity of fires between or
among sites.  Two situations can be tested, depending on the research
objective.  First, the synchroneity of fires can be tested between two
subsets of samples from within the same site.  For example, a single,
contiguous site may be divided into an upper elevation site and a lower
elevation site, and tests performed to determine whether the fire years
are synchronous between these two subsets.  There should be some prior
determination by the researcher of the criterion for dividing the site
into two subsites (e.g. topographic effects, soil type, habitat type,
land use history, etc.).  Second, the synchroneity of fires can be tested
between two non-adjacent sites under the null hypothesis that the fire
years that occurred at both sites are not associated.  For example, two
fire history sites may be separated by 50 kilometers, and the researcher
has observed some fire and non-fire years are common between the two
sites.  If the synchroneity of fire years between these two sites is
indeed significant, then the researcher can postulate which external
mechanism(s) may be responsible for causing this synchroneity of fire and
n on-fire years between sites separated by so great a distance.

If two separate sites are to be investigated, the researcher has two
options: (1) combine the individual samples from both sites into one data
file using the Merge Matrices routine in the Data Entry Module; or (2)
conduct the analyses only on the composite information for both sites by
exporting this information to a data file in the Composite Axis Menu of
the Graphics Module.  Both options are suitable, but the latter is much
faster because only two time series are analyzed as opposed to the many
individual series that make up each composite series.  If the first
option is chosen, the user must first divide the site after the
appropriate series.  If the second option is chosen, the software will
automatically divide the data file after the first composite series.  To
begin testing for synchroneity, select "Y" from the menu, then run the
analyses by selecting "G".

The output produced by this option consists of five parts: (1) summary
information on the synchroneity of fires between the two sites; (2) a
2 X 2 contingency analysis using the chi-squared test; (3) a 2 X 1
contingency analysis also using the chi-squared test; (4) summary
information for various similarity indices; and (5) results from the
Wald-Wolfowitz runs test. The fire years for both sites are listed in the
first two columns, while columns three and four provide information on
the synchroneity of fires.  The last column provides information on the
number of synchronous or asynchronous runs that occurred over time
between the two sites.  In this column, a "+" sign always indicates a
synchronous event, while a "-" sign indicates an asynchronous event.

The first two tests performed are chi-squared tests of association.  The
first of these considers all years from the beginning to the end of the
range of years specified by the user.  A 2 X 2 analysis consists of
four cells.  The first cell ("a") contains the number of "++" fire events
in which fire occurred at both sites.  The second cell ("b") contains the
number of events that occurred at Site A, but not at Site B ("+-").  The
third cell ("c") contains the number events that occurred at Site B, but
not at Site A ("-+").  Finally, the fourth cell ("d") contains the number
of years in which fire did not occur at either site ("--").  These values
are printed in column two, while column three contains the expected
number of events under the null hypothesis that fire years between the
two sites are not associated.  The fourth column contains the calculation
of the chi-squared statistic.  Significance is denoted at the .05, .01,
and .005 levels.  If the expected value in any one cell falls below 1.0,
then the Yates correction for continuity is applied, and the corrected
chi-squared is printed as well.

The 2 X 1 contingency analysis expands on this first analysis.  Despite
the fact that 33 fire years were indeed synchronous between the two
sites, there were still 14 years in which fire occurred at only one of
the two sites.  This analysis is similar to a coin toss in which the
results can be either synchronous fire years ("heads") or non-synchronous
fire years ("tails").  Only fire years are considered in this analysis.
The number of fire years in which fire did not occur at either site (the
"d" cell) is not used.  In addition, note that the values in the "b" and
"c" cells are grouped because either situation is a non-synchronous fire
year.  Because only fire years are used in this analysis, statistical
significance is more difficult to achieve due to the fewer observations
and cells used.  If the two subsites are contiguous, as in this
situation, then we would expect fires to have spread between the two
subsites, and the chi-squared statistic should indeed be significant.  If
the statistic is not significant (indicated by "NS" in the output), then
the researcher should postulate possible reasons that would prevent fire
from spreading between the two subsites.

The fourth section of the output lists various similarity indices
commonly used in ecological analyses.  Similarity indices measure the
degree of association between two sets of binary (presence-absence, or
plus-minus) data.  These indices also use the exact same values for the
four cells as used in the calculation of the chi-squared statistic, which
itself is also a measure of association.  Most indices, however, do not
use the value for the "d" cell, because the researcher is usually not
interested in the complete absence of data.  The first four indices
listed, the Ochiai, Jaccard, Dice, and Kulczynski indices, do not use the
"d" cell.  Not incorporating the "d" cell into the analyses has spurred
continuous debate in the ecological literature, because some researchers
argue that the complete absence of data provides some form of informa-
tion.  For this reason, two additional indices are listed that incor-
porate the "d" cell into their calculations.  These are the Yule Index
(also known as the "tetrachoric correlation coefficient") and the
Baroni-Urbani & Buser Index.  Previous studies have determined that
these six indices were the most appropriate of 43 indices evaluated based
on certain criteria.

Similarity indices usually have a range between zero, indicating no
association whatsoever between the two data sets, and one, indicating
perfect association between the two data sets.  (The Yule Index has a
range between -1 and 1, similar to r-values, or correlation coefficients. 
The values have been rescaled to range between zero and one to maintain
compatibility among indices

The last test performed is the Wald-Wolfowitz runs test, and is similar
to the previous tests in that only fire years are considered in this
analysis.  The sequence of synchronous and asynchronous fire years forms
separate sequences of positive and negative runs.  These runs can be
statistically evaluated to determine whether the overall sequence was
generated by a random process.  If the total number of runs is less than
that expected due simply by chance, then the sequence of runs was not
generated by a random process.

Note that rejection of the null hypothesis can actually lead to two
alternate hypotheses.  If the sequence was not generated by a random
process, then two situations can arise: either there were more negative
(non-synchronous runs) or more positive (synchronous) runs than expected
simply by chance.  The "direction" of the alternate hypothesis is
determined by the larger of the positive or negative observations.  The
test statistic used is the W-statistic, which is the expected number of
runs plus two standard deviations, both of which are printed in the
output.  If the total number of runs is less than this statistic, then
the sequence of runs was most likely not generated by a random process.
Only the .05 level is used in this analysis.
