RATIO OF STANDARD DEVIATIONS CONFIDENCE LIMITS
RATIO OF VARIANCES CONFIDENCE LIMITS

RATIO OF STANDARD DEVIATIONS CONFIDENCE LIMITS
RATIO OF VARIANCES CONFIDENCE LIMITS

Analysis Command

Generates a confidence interval for the ratio of two standard deviations (or variances).

This is the confidence interval for the ratio of the standard deviations (or variances) for two unpaired samples. If the confidence interval for the ratio, this implies that the standard deviations (or variances) are equal.

The confidence interval or the ratio of the variances is:

\( \left( \frac{s_{1}^{2}/s_{2}^{2}} {F(\alpha2,\nu_{2}-1,\nu_{1}-1)}, (s_{1}^{2}/s_{2}^{2}) F(\alpha2,\nu_{2}-1,\nu_{1}-1) \right) \)

where \( s_{1}^{2} \) and \( s_{2}^{2} \) are the sample variances, \( n_1 \) and \( n_2 \) are the sample sizes and F is the percent point function of the F distribution. The confidence intervals are given for several values of alpha. This interval is not necessarily symmetric about the ratio \( \frac{s_{1}^{2}} {s_{2}^{2}} \).

To obtain the confidence interval for the ratio of standard deviations, take the square root of the lower and upper confidence limits for the ratio of the variances.

This confidence interval assumes that both distributions have normal distributions. It is also known that this interval is quite senstive to departures from normality. There have been several proposals in the literature to make this test more robust. This is discussed further in the "Note:" section below.

RATIO OF STANDARD DEVIATIONS CONFIDENCE LIMITS <y1> <y2>
            <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

RATIO OF VARIANCES CONFIDENCE LIMITS <y1> <y2>
            <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

RATIO OF STANDARD DEVIATIONS CONFIDENCE LIMITS Y1 Y2
RATIO OF SD CONFIDENCE LIMITS Y1 Y2
RATIO OF VARIANCE CONFIDENCE LIMITS Y1 Y2
RATIO OF SD CONFIDENCE LIMITS Y1 Y2 SUBSET Y2 > 0

This ratio is known to be quite sensitive to departures from normality.

Shoemaker (2003) proposed the following adjustment to the degrees of freedom to make the F test more robust against non-normality.

The degrees of freedom are

\( r_{i} = \frac{2 n_{i}} {\frac{\mu_{4}}{\sigma^{4}} - \frac{n_{i}-3}{n_{i}-1}} \)

where

\( \mu_{4} = \frac{\sum_{i=1}^{2}{\sum_{j=1}^{n_i} {(y_{ij} - \bar{y}_{i})^{4}}}} {n_{1} + n_{2}} \)

\( \sigma^{2} = \frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}} {n_{1} + n_{2}} \)

with \( y_{ij} \), \( n_{i} \), \( s_{i} \) and \( \bar{y}_{i} \) denoting the data, the sample size, the standard deviation and the mean, respectively, of the i-th variable.

This typically results in fractional degrees of freedom. The Dataplot FCDF, FPDF and FPPF routines were updated to support fractional degrees of freedom (fractional degrees of freedom less than 1 are set to 1).

The justification and derivation of this modification is given in Shoemaker's paper. Simulations by Shoemaker indicated that the standard F test tends to be too conservative for light tailed distributions and too liberal for heavy tailed and skewed distributions. Using the Shoemaker modifications for degrees of freedom generally had good coverage properties with the exception of heavily skewed distributions with small samples where it was still too liberal.

Bonett (2006) suggested that rounding the numerator degrees of freedom up to the nearest integer degrees of freedom and rounding the denominator degrees of freedom down to the nearest integer degrees of freedom (but not less than 1) prevented some anomalous behavior while reducing coverage performance only slightly.

Bonett proposed the following interval:

\( \exp{\left( \ln{(c\frac{s_{1}^{2}}{s_{2}^{2}})} \pm \mbox{se} \Phi^{-1}(\alpha/2) \right) } \)

where

\( \gamma_{4} \)	=	\( (n_1 + n_2) \frac{ \sum_{i=1}^{2}{\sum_{j=1}^{n_i}{(y_{ij} - m_{i})^{4}}}} {(\sum_{i=1}^{2}{\sum_{j=1}^{n_i}{(y_{ij} - \mu_{i})^{2}}} )^{2}} \)
\( m_{i} \)	=	sample trimmed mean of the i-th variable with trimming proportion \( \frac{1}{\sqrt{2 (n{i} - 4)}} \)
\( \bar{y_{i}} \)	=	sample mean of the i-th variable
se	=	\( \sqrt{ \frac{\gamma_{4} - k_{1}} {n_{1} - 1} + \frac{\gamma_{4} - k_{2}} {n_{2} - 1} } \)
\( k_1 \)	=	\( \frac{n_{1}-3}{n_{1}} \)
\( k_2 \)	=	\( \frac{n_{2}-3}{n_{2}} \)
c	=	\( \frac{\frac{n_{1}} {n_{1} - \Phi^{-1}(\alpha/2)}} {\frac{n_{2}} {n_{2} - \Phi^{-1}(\alpha/2)}} \)
\( \Phi^{-1} \)	=	the percent point function of the normal distribution

The parameter c is a small sample adjustment to help equalize the tail probabilities. It equals 1 when n1 and n2 equal and approaches 1 as n1 and n2 get large.

The above formula is for the variance. For the standard deviation, take the square root.

This interval is based on a standard normal distribution rather than an F distribution.

The justification and derivation of this test is given in the Bonett paper. Based on his simulations, he claims this method improves somewhat on the Shoemaker modification for heavy-tailed (particularly skewed) distributions.

To specify Shoemaker's modification with fractional degrees of freedom, enter

SET F TEST DEGREES OF FREEDOM SHOEMAKER

To specify Shoemaker's modification with integral degrees of freedom as suggested by Bonett, enter

SET F TEST DEGREES OF FREEDOM SHOEMAKER ROUNDED

To specify Bonett's method, enter

SET F TEST DEGREES OF FREEDOM BONETT

Shoemaker suggests that when more than two groups of data are available, it can improve accuracy to use all of the groups in the estimates of the pooled location (mu4) and pooled variance (sigma**2) instead of just the two groups being tested. That is,
\( \mu_{4} = \frac{\sum_{i=1}^{k}{\sum_{j=1}^{n_i} {(y_{ij} - \bar{y}_{i})^{4}}}} {n_{1} + n_{2} + ... + n_{k}} \)

\( \sigma^{2} = \frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2} + ... + (n_{k} - 1) s_{k}} {n_{1} + n_{2} + ... + n_{k}} \)

To use these values, enter the commands

SET SHOEMAKER F TEST POOLED MU <value>
SET SHOEMAKER F TEST POOLED VARIANCE <value>


To reset the default of Dataplot computing these from the two groups being tested, enter

SET SHOEMAKER F TEST POOLED MU
SET SHOEMAKER F TEST POOLED VARIANCE 0

The following parameters are saved after the F test is performed.

CUTLOW90	-	lower 90% confidence limit
CUTUPP90	-	upper 90% confidence limit
CUTLOW95	-	lower 95% confidence limit
CUTUPP95	-	upper 95% confidence limit
CUTLOW99	-	lower 99% confidence limit
CUTUPP99	-	upper 99% confidence limit
CUTLW999	-	lower 99.5% confidence limit
CUTUP999	-	upper 99.5% confidence limit

In addition to the RATIO OF STANDARD DEVIATION CONFIDENCE LIMITS command, the following commands can also be used:
LET LCL = RATIO OF SD LOWER CONFIDENCE LIMTIS Y1 Y2
LET UCL = RATIO OF SD UPPER CONFIDENCE LIMTIS Y1 Y2
LET LCL = RATIO OF VARIANCE LOWER CONFIDENCE LIMTIS Y1 Y2
LET UCL = RATIO OF VARIANCE UPPER CONFIDENCE LIMTIS Y1 Y2


In addition to the above LET commands, built-in statistics are supported for 30+ different commands (enter HELP STATISTICS for details).

The F TEST can be used to generate the corresponding hypothesis test.

None

SD is a synonym for STANDARD DEVIATION
CONFIDENCE INTERVAL is a synonym for CONFIDENCE LIMIT

F TEST	=	Perform a normal-based test that two standard deviations are equal.
SIEGEL TUKEY TEST	=	Perform a ranks based test that two standard deviations are equal.
BARTLETT TEST	=	Perform a k-sample Bartlett test for homogeneous variances.
LEVENE TEST	=	Perform a k-sample Levene test for homogeneous variances.
SQUARED RANKS TEST	=	Perform a k-sample squared ranks test for homogeneous variances.
KLOTZ TEST	=	Perform a k-sample Klotz test for homogeneous variances.
SD CONFIDENCE LIMITS	=	Compute the confidence limits for the standard deviation.
CHI-SQUARE TEST	=	Performs a one sample chi-square test that the standard deviation is equal to a given value.
T TEST	=	Performs a two-sample t test for equal means.
CONFIDENCE LIMITS	=	Compute the confidence limits for the mean of a sample.
STANDARD DEVIATION	=	Computes the standard deviation of a variable.

F-based intervals for the ratio of two standard deviations are discussed in most introductory statistics books.

Shoemaker (2003), "Fixing the F-Test for Equal Variances," The American Statistician, Vol. 57, pp. 105-114.

Bonett (2006), "Robust Confidence Intervals for a Ratio of Standard Deviations," Applied Pyschological Measurement, Vol. 30, No. 5, pp. 432-439.

Confirmatory Data Analysis

2023/06

 
    
 
SKIP 25
READ AUTO83B.DAT Y1 Y2
DELETE Y2 SUBSET Y2 < 0
.
RATIO OF SD CONFIDENCE LIMITS Y1 Y2
    
    The following output is generated
    
    
             Confidence Limits for the Ratio of Standard Deviations
  
 Numerator Variable:   Y1
 Denominator Variable: Y2
  
  
 Summary Statistics for Numerator Variable:
 Number of Observations:                               249
 Sample Mean:                                     20.14458
 Sample Standard Deviation:                        6.41470
  
 Summary Statistics for Variable 2:
 Number of Observations:                                79
 Sample Mean:                                     30.48101
 Sample Standard Deviation:                        6.10771
  
 Degrees of Freedom (Numerator):                       248
 Degrees of Freedom (Denomerator):                      78
 Ratio of Standard Deviations:                     1.05026
  
  
 --------------------------------------------------
   Confidence                  Lower          Upper
    Value (%)   Ratio          Limit          Limit
 --------------------------------------------------
       50.000   1.050        0.99058        1.12161
       80.000   1.050        0.93827        1.18830
       90.000   1.050        0.90869        1.23086
       95.000   1.050        0.88401        1.26950
       99.000   1.050        0.83820        1.35006
       99.900   1.050        0.78873        1.45285
    
    
    

 
. Step 1:   Read the data
.
skip 25
read shoemake.dat bottom mid surface
let y x = stack bottom mid surface
let temp1 = cross tabulate mean y x
let temp2 = (y - temp1)**4
let num = sum temp2
let den = size y
let mu4 = num/den
let y1var = variance bottom
let y2var = variance mid
let y3var = variance surface
let n1 = size bottom
let n2 = size mid
let n3 = size surface
let num   = (n1 - 1)*y1var + (n2 -1)*y2var + (n3 - 1)*y3var
let poolvar = num/den
.
. Step 2:   Generate a box plot
.
character box plot
line box plot
xlimits 1 3
major xtic mark number 3
minor xtic mark number 0
xtic mark offset 0.5 0.5
x1tic mark label format alpha
x1tic mark label content Bottom Middepth Surface
tic mark label case asis
set box plot fences on
box plot y x



.
. Step 3:   Default confidence intercal
.
set f test degrees of freedom default
print "Standard F interval"
ratio of sd confidence limits bottom mid
let lcl = ratio of sd lower confidence limit bottom mid
let ucl = ratio of sd upper confidence limit bottom mid
print lcl ucl
    

The following output is generated

Standard F interval
  
             Confidence Limits for the Ratio of Standard Deviations
  
 Numerator Variable:   BOTTOM
 Denominator Variable: MID
  
  
 Summary Statistics for Numerator Variable:
 Number of Observations:                                10
 Sample Mean:                                      6.02100
 Sample Standard Deviation:                        1.58184
  
 Summary Statistics for Variable 2:
 Number of Observations:                                10
 Sample Mean:                                      5.01900
 Sample Standard Deviation:                        1.10440
  
 Degrees of Freedom (Numerator):                         9
 Degrees of Freedom (Denomerator):                       9
 Ratio of Standard Deviations:                     1.43231
  
  
 --------------------------------------------------
   Confidence                  Lower          Upper
    Value (%)   Ratio          Limit          Limit
 --------------------------------------------------
       50.000   1.432        1.13558        1.80659
       80.000   1.432        0.91688        2.23750
       90.000   1.432        0.80334        2.55373
       95.000   1.432        0.71384        2.87392
       99.000   1.432        0.56003        3.66322
       99.900   1.432        0.41217        4.97734
  
  
 THE COMPUTED VALUE OF THE CONSTANT LCL           =   0.7138401
  
  
 THE COMPUTED VALUE OF THE CONSTANT UCL           =    2.873916
  

 PARAMETERS AND CONSTANTS--

    LCL     --        0.71384
    UCL     --        2.87392
    

.
. Step 4:   Shoemaker F-test
.
set f test degrees of freedom shoemaker
print "Shoemaker degrees of freedom interval"
ratio of sd confidence limits bottom mid
print "Use pooled location and variance for Shoemaker"
set shoemaker f test pooled variance poolvar
set shoemaker f test pooled mu       mu4
ratio of sd confidence limits bottom mid
print "Shoemaker rounded degrees of freedom F Test"
set f test degrees of freedom shoemaker rounded
ratio of sd confidence limits bottom mid
set shoemaker f test pooled variance 0
set shoemaker f test pooled mu
    The following output is generated
    
    
Shoemaker degrees of freedom interval
  
             Confidence Limits for the Ratio of Standard Deviations
              (Use Shoemaker Modifications to Degrees of Freedom)
  
 Numerator Variable:   BOTTOM
 Denominator Variable: MID
  
  
 Summary Statistics for Numerator Variable:
 Number of Observations:                                10
 Sample Mean:                                      6.02100
 Sample Standard Deviation:                        1.58184
  
 Summary Statistics for Variable 2:
 Number of Observations:                                10
 Sample Mean:                                      5.01900
 Sample Standard Deviation:                        1.10440
  
 Degrees of Freedom (Numerator):                  12.42315
 Degrees of Freedom (Denomerator):                12.42315
 Ratio of Standard Deviations:                     1.43231
  
  
 --------------------------------------------------
   Confidence                  Lower          Upper
    Value (%)   Ratio          Limit          Limit
 --------------------------------------------------
       50.000   1.432        1.17755        1.74220
       80.000   1.432        0.98426        2.08433
       90.000   1.432        0.88188        2.32631
       95.000   1.432        0.80006        2.56420
       99.000   1.432        0.65671        3.12394
       99.900   1.432        0.51437        3.98837
  
Use pooled location and variance for Shoemaker
  
 THE FORTRAN COMMON SCALAR SHOEF    HAS JUST BEEN SET TO   0.1251562E+01
  
 THE FORTRAN COMMON SCALAR SHOEF    HAS JUST BEEN SET TO   0.4585221E+01
  
             Confidence Limits for the Ratio of Standard Deviations
              (Use Shoemaker Modifications to Degrees of Freedom)
  
 Numerator Variable:   BOTTOM
 Denominator Variable: MID
  
  
 Summary Statistics for Numerator Variable:
 Number of Observations:                                10
 Sample Mean:                                      6.02100
 Sample Standard Deviation:                        1.58184
  
 Summary Statistics for Variable 2:
 Number of Observations:                                10
 Sample Mean:                                      5.01900
 Sample Standard Deviation:                        1.10440
  
 Degrees of Freedom (Numerator):                   9.30474
 Degrees of Freedom (Denomerator):                 9.30474
 Ratio of Standard Deviations:                     1.43231
  
  
 --------------------------------------------------
   Confidence                  Lower          Upper
    Value (%)   Ratio          Limit          Limit
 --------------------------------------------------
       50.000   1.432        1.14021        1.79924
       80.000   1.432        0.92423        2.21969
       90.000   1.432        0.81186        2.52695
       95.000   1.432        0.72313        2.83698
       99.000   1.432        0.57032        3.59714
       99.900   1.432        0.42285        4.85169
  
Shoemaker rounded degrees of freedom F Test
  
 THE FORTRAN COMMON CHARACTER VARIABLE F   TEST HAS JUST BEEN SET TO SHO2
  
             Confidence Limits for the Ratio of Standard Deviations
             (Shoemaker Modifications with Rounded Degrees of Freedom)
  
 Numerator Variable:   BOTTOM
 Denominator Variable: MID
  
  
 Summary Statistics for Numerator Variable:
 Number of Observations:                                10
 Sample Mean:                                      6.02100
 Sample Standard Deviation:                        1.58184
  
 Summary Statistics for Variable 2:
 Number of Observations:                                10
 Sample Mean:                                      5.01900
 Sample Standard Deviation:                        1.10440
  
 Degrees of Freedom (Numerator):                        10
 Degrees of Freedom (Denomerator):                       9
 Ratio of Standard Deviations:                     1.43231
  
  
 --------------------------------------------------
   Confidence                  Lower          Upper
    Value (%)   Ratio          Limit          Limit
 --------------------------------------------------
       50.000   1.432        1.31822        3.25440
       80.000   1.432        0.87399        4.95711
       90.000   1.432        0.67922        6.43618
       95.000   1.432        0.54288        8.13193
       99.000   1.432        0.34378       13.16491
       99.900   1.432        0.19419       24.23791
  
  
 THE FORTRAN COMMON SCALAR SHOEF    HAS JUST BEEN SET TO   0.0000000E+00
  
 THE FORTRAN COMMON SCALAR SHOEF    HAS JUST BEEN SET TO  -0.1000000E+01
    
    
.
. Step 5:   Bonett
.
set f test degrees of freedom bonett
print "Bonett method"
ratio of sd confidence limits bottom mid
    The following output is generated
    
    
Bonett method
  
             Confidence Limits for the Ratio of Standard Deviations
                         (Bonett Method for Robustness)
  
 Numerator Variable:   BOTTOM
 Denominator Variable: MID
  
  
 Summary Statistics for Numerator Variable:
 Number of Observations:                                10
 Sample Mean:                                      6.02100
 Sample Standard Deviation:                        1.58184
  
 Summary Statistics for Variable 2:
 Number of Observations:                                10
 Sample Mean:                                      5.01900
 Sample Standard Deviation:                        1.10440
  
 Ratio of Standard Deviations:                     1.43231
  
  
 --------------------------------------------------
   Confidence                  Lower          Upper
    Value (%)   Ratio          Limit          Limit
 --------------------------------------------------
       50.000   1.432        1.28002        1.56990
       80.000   1.432        1.12547        1.68414
       90.000   1.432        1.02186        1.74895
       95.000   1.432        0.92262        1.80328
       99.000   1.432        0.68853        1.90498
       99.900   1.432        0.19075        2.01659