RATIO OF MEANS CONFIDENCE INTERVAL

RATIO OF MEANS CONFIDENCE INTERVAL

Analysis Command

Generates a confidence interval for the ratio of two means for paired samples.

There are cases where a measurement is actually the ratio of two different measurements. That is,
\[ R_i = \frac{Y_i} {X_i} \]

It is often desired to generate the confidence interval for this ratio. Note that computing a standard confidence interval for R does not generate satisfactory results. This is due to the fact that, assuming Y and X are independent,

\[ \begin{array}{lcl} E[\hat{R}] & = & E[\frac{\hat{Y}}{\hat{X}}] \\ & = & E[\hat{Y}] E[1/\hat{X}] \end{array} \]

However, \( E[1/\hat{X}] \) is not equal to \( 1/E[\hat{X}] \).

There have been a number of approaches to this problem. This command supports three different methods.

Fieller derived confidence intervals for the case where Y and X are distributed as bivariate normal. Define the quantities

\( \bar{X} \)	=	mean of X
\( \bar{Y} \)	=	mean of Y
\( \hat{\sigma}_{\bar{X}}^{2} \)	=	variance of \( \bar{X} \)
	=	\( \frac{1}{N(N-1)} \sum_{i=1}^{N}{(X_{i} - \bar{X})^2} \)
\( \hat{\sigma}_{\bar{Y}}^{2} \)	=	variance of \( \bar{Y} \)
	=	\( \frac{1}{N(N-1)} \sum_{i=1}^{N}{(Y_{i} - \bar{Y})^2} \)
\( \hat{\sigma}_{\bar{X},\bar{Y}} \)	=	\( \frac{1}{N(N-1)} \sum_{i=1}^{N} {(X_i - \bar{X})(Y_i - \bar{Y})} \)
tq	=	t percent point value with N - 1 degrees of freedom

The test statistic is

\[ \hat{R} = \frac{\bar{Y}} {\hat{X}} \]

For Fieller's confidence limits, we first compute

\[ \frac{\bar{X}^2} {\hat{\sigma}_{\bar{X}}^2} \]

If this quantity is less than or equal to \( t_{q}^2 \) then an unbounded interval results and Dataplot will not generate the confidence interval. Basically, this results if the confidence interval for X contains zero.

If this quantity is less than or equal to \( t_{q}^2 \) then the following confidence interval is obtained

\( \mbox{Lower Limit} = \frac{(\bar{X} \bar{Y} - t_{q}^{2} \hat{\sigma}_{\bar{X} \bar{Y}}) - \sqrt{ (\bar{X} \bar{Y} - t_{q}^{2} \hat{\sigma}_{\bar{X} \bar{Y}})^2 - (\bar{X}^2 - t_{q}^{2} \hat{\sigma}_{\bar{X}}^{2}) (\bar{Y}^2 - t_{q}^{2} \hat{\sigma}_{\bar{Y}}^{2})}} {\bar{X}^2 - t_{q}^{2} \hat{\sigma}_{\bar{X}}^{2}} \)

\( \mbox{Upper Limit} = \frac{(\bar{X} \bar{Y} - t_{q}^{2} \hat{\sigma}_{\bar{X} \bar{Y}}) + \sqrt{ (\bar{X} \bar{Y} - t_{q}^{2} \hat{\sigma}_{\bar{X} \bar{Y}})^2 - (\bar{X}^2 - t_{q}^{2} \hat{\sigma}_{\bar{X}}^{2}) (\bar{Y}^2 - t_{q}^{2} \hat{\sigma}_{\bar{Y}}^{2})}} {\bar{X}^2 - t_{q}^{2} \hat{\sigma}_{\bar{X}}^{2}} \)

The large sample approximation method (this is called the Taylor or delta method in the Franz paper) generates the following confidence interval

\( \mbox{Lower Limit} = \hat{R} - t_{(\alpha/2,n-1)} \hat{R} \sqrt{C_{\bar{Y}\bar{Y}} + C_{\bar{X}\bar{X}} - 2 C_{\bar{Y}\bar{X}}} \)

\( \mbox{Upper Limit} = \hat{R} + t_{(\alpha/2,n-1)} \hat{R} \sqrt{C_{\bar{Y}\bar{Y}} + C_{\bar{X}\bar{X}} - 2 C_{\bar{Y}\bar{X}}} \)

where

\( C_{\bar{Y},\bar{Y}} \)	=	\( \frac{s_{Y}^{2}} {N \bar{Y}^2} \)
\( C_{\bar{X},\bar{X}} \)	=	\( \frac{s_{X}^{2}} {N \bar{X}^2} \)
\( C_{\bar{Y},\bar{X}} \)	=	\( \frac{s_{X Y}} {N \bar{X} \bar{Y}} \)

with \( s_{Y} \), \( s_{X} \), and \( s_{X Y} \) denoting the standard deviation of Y, the standard deviation of X, and the covariance between X and Y, respectively.

The log ratio method generates the following confidence interval

\( \mbox{Lower Limit} = \hat{R} \exp{(-t_{(\alpha/2,n-1)} \sqrt{C_{\bar{Y} \bar{Y}} + C_{\bar{X} \bar{X}} - 2 C_{\bar{Y} \bar{X}}})} \)

\( \mbox{Upper Limit} = \hat{R} \exp{(t_{(\alpha/2,n-1)} \sqrt{C_{\bar{Y} \bar{Y}} + C_{\bar{X} \bar{X}} - 2 C_{\bar{Y} \bar{X}}})} \)

The large sample approximation and the log ratio method do not generate unbounded intervals. Also, the log ratio method can generate asymmetric intervals.

Note that there is some disagreement in the literature about the appropriateness of these methods. For example, Franz argues that the unbounded intervals are a result of the denominator being close to zero with the consequence that the ratio can assume arbitrarily large values. Therefore any method that does not allow for unbounded intervals is not valid. On the other hand, Sherman argues that the unbounded Fieler intervals are simply nonsensical and advocates the use of the large sample approximation and log ratio methods.

To specify the method to use, enter the command

SET RATIO OF MEANS METHOD <FIELER/LOG RATIO/LARGE SAMPLE>

RATIO OF MEANS CONFIDENCE INTERVAL <y1> <y2>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first (numerator) response variable;
            <y2> is the second (denominator) response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

The variables <y1> and <y2> must be of the same length and are assumed to be paired.

RATIO OF MEANS CONFIDENCE INTERVAL Y X
RATIO OF MEANS CONFIDENCE INTERVAL Y X SUBSET TAG > 2
RATIO OF MEANS CONFIDENCE INTERVAL Y1 Y2 SUBSET Y1 > 0

A table of confidence intervals is printed for alpha levels of 50.0, 75.0, 90.0, 95.0, 99.0, 99.9, 99.99, and 99.999. The sample sizes, sample means, sample standard deviations, and the standard error are also printed. The t-value and t-value X standard error are printed in the table.

The RATIO OF MEANS CONFIDENCE LIMIT command automatically saves the following parameters:

CUTLOW90	=	the lower 90% confidence limit
CUTUPP90	=	the upper 90% confidence limit
CUTLOW95	=	the lower 95% confidence limit
CUTUPP95	=	the upper 95% confidence limit
CUTLOW99	=	the lower 99% confidence limit
CUTUPP99	=	the upper 99% confidence limit
CTLOW999	=	the lower 99.9% confidence limit
CTUPP999	=	the upper 99.9% confidence limit

In addition to the RATIO OF MEANS CONFIDENCE LIMIT command, the following commands can also be used:
LET A = RATIO OF MEANS Y X


LET ALPHA = <value>
LET A = RATIO OF MEANS LOWER CONFIDENCE LIMIT Y X
LET A = RATIO OF MEANS UPPER CONFIDENCE LIMIT Y X


These statistics can be used in a number of commands. For details, enter

HELP STATISTICS

The default method is Fieler's method.

None

CONFIDENCE LIMITS	=	Generate a confidence interval for the mean.
DIFFERENCE OF MEANS CONFIDENCE LIMIT	=	Generate a confidence interval for the difference of two means.
T-TEST	=	= Perform a two sample t-test.
BIHISTOGRAM	=	Generate a bihistogram.
QUANTILE-QUANTILE PLOT	=	= Generate a quantile-quantile plot.
KOLMOGOROV-SMIRNOV TWO SAMPLE TEST	=	Generate a Kolmogorov- Smirnov two sample test.

V. H. Franz ((2007), "Ratios: A Short Guide to Confidence Limits and Proper Use," arXiv:0710.2024 [stat.AP].

E. C. Fieler (1940), "The Biological Standardization of Insulin," Supplement to the Journal of the Royal Statistical Society, Vol. 7, No. 1, pp. 1-64.

E. C. Fieler (1940), "A Fundamental Formula in the Statistics of Biological Assays and Some Applications", Quarterly Journal of Pharmacy and Pharmacology, Vol. 17, pp. 117-123.

E. C. Fieler (1940), "Some Problems in Interval Estimation," Journal of the Royal Statistical Society (B), Vol. 16, No. 2, pp. 175-185.

Sherman, Maity, and Wang (2011), "Inferences for the Ratio: Fieller's Interval, Log Ratio, and Large Sample Based Confidence Intervals", AStA Adv Stat Anal 95:313–323.

Cochran (1977), "Sampling Techniques," Wiley, New York.

Lohr (2009), "Sampling: Design and Analysis," Second Edition, Brooks/Cole, Pacific Grove.

Confirmatory Data Analysis

2019/09

 
. Step 1:   Define data (taken from Sherman article, original source
.
.           Lehtonen and Pahkinen (2004), "Practical Methods for
.           Design and Analysis of Complex Surveys," 2nd Edition,
.           New York: Wiley.
.
read y x
 4123   26881
  760    4896
  721    3730
  142     556
  187    1463
  331    1946
  127     834
  219     932
end of data
.
. Step 2:   Large sample interval
.
let alpha = 0.95
set write decimals 4
set ratio of means method large sample
let r1   = ratio of means y x
let r1ll = ratio of means lower confidence limit y x
let r1ul = ratio of means upper confidence limit y x
.
ratio of means confidence limit y x
pause
    

The following output is generated

 PARAMETERS AND CONSTANTS--

    R1      --         0.1603
    R1LL    --         0.1452
    R1UL    --         0.1754
 
            Confidence Limits for the Ratio of Means
              (Large Sample Approximation Method)
 
Numerator Variable:   Y
Denominator Variable: X
 
 
Summary Statistics for Numerator Variable:
Number of Observations:                                 8
Sample Mean:                                     826.2500
Sample Standard Deviation:                      1355.6149
Sample Coefficient of Variation:                   1.6407
 
Summary Statistics for Variable 2:
Number of Observations:                                 8
Sample Mean:                                    5154.7500
Sample Standard Deviation:                      8909.8733
Sample Coefficient of Variation:                   1.7285
 
Correlation:                                       0.9991
 
 
 
---------------------------------------------------------
  Confidence                         Lower          Upper
   Value (%)          Ratio          Limit          Limit
---------------------------------------------------------
      50.000         0.1603         0.1558         0.1648
      75.000         0.1603         0.1523         0.1683
      90.000         0.1603         0.1482         0.1724
      95.000         0.1603         0.1452         0.1754
      99.000         0.1603         0.1380         0.1826
      99.900         0.1603         0.1258         0.1948
      99.990         0.1603         0.1102         0.2104
      99.999         0.1603         0.0895         0.2311
    

. set ratio of means method log ratio let r2 = ratio of means y x let r2ll = ratio of means lower confidence limit y x let r2ul = ratio of means upper confidence limit y x . ratio of means confidence limit y x pause The following output is generated

 PARAMETERS AND CONSTANTS--

    R2      --         0.1603
    R2LL    --         0.1459
    R2UL    --         0.1761
 
            Confidence Limits for the Ratio of Means
                       (Log Ratio Method)
 
Numerator Variable:   Y
Denominator Variable: X
 
 
Summary Statistics for Numerator Variable:
Number of Observations:                                 8
Sample Mean:                                     826.2500
Sample Standard Deviation:                      1355.6149
Sample Coefficient of Variation:                   1.6407
 
Summary Statistics for Variable 2:
Number of Observations:                                 8
Sample Mean:                                    5154.7500
Sample Standard Deviation:                      8909.8733
Sample Coefficient of Variation:                   1.7285
 
Correlation:                                       0.9991
 
 
 
---------------------------------------------------------
  Confidence                         Lower          Upper
   Value (%)          Ratio          Limit          Limit
---------------------------------------------------------
      50.000         0.1603         0.1558         0.1649
      75.000         0.1603         0.1525         0.1685
      90.000         0.1603         0.1486         0.1728
      95.000         0.1603         0.1459         0.1761
      99.000         0.1603         0.1394         0.1842
      99.900         0.1603         0.1293         0.1987
      99.990         0.1603         0.1173         0.2191
      99.999         0.1603         0.1031         0.2493
    

. set ratio of means method fieler let r3 = ratio of means y x let r3ll = ratio of means lower confidence limit y x let r3ul = ratio of means upper confidence limit y x . ratio of means confidence limit y x The following output is generated

 PARAMETERS AND CONSTANTS--

    R3      --         0.1603
    R3LL    --***************
    R3UL    --***************
 
***** WARNING: RATIO OF MEANS CONFIDENCE LIMITS--
      FOR ALPHA (   90.00000    ), THE FIELLER INTERVAL IS UNBOUNDED.
 
***** WARNING: RATIO OF MEANS CONFIDENCE LIMITS--
      FOR ALPHA (   95.00000    ), THE FIELLER INTERVAL IS UNBOUNDED.
 
***** WARNING: RATIO OF MEANS CONFIDENCE LIMITS--
      FOR ALPHA (   99.00000    ), THE FIELLER INTERVAL IS UNBOUNDED.
 
***** WARNING: RATIO OF MEANS CONFIDENCE LIMITS--
      FOR ALPHA (   99.90000    ), THE FIELLER INTERVAL IS UNBOUNDED.
 
***** WARNING: RATIO OF MEANS CONFIDENCE LIMITS--
      FOR ALPHA (   99.99000    ), THE FIELLER INTERVAL IS UNBOUNDED.
 
***** WARNING: RATIO OF MEANS CONFIDENCE LIMITS--
      FOR ALPHA (   99.99900    ), THE FIELLER INTERVAL IS UNBOUNDED.
 
            Confidence Limits for the Ratio of Means
                        (Fieller Method)
 
Numerator Variable:   Y
Denominator Variable: X
 
 
Summary Statistics for Numerator Variable:
Number of Observations:                                 8
Sample Mean:                                     826.2500
Sample Standard Deviation:                      1355.6149
Sample Coefficient of Variation:                   1.6407
 
Summary Statistics for Variable 2:
Number of Observations:                                 8
Sample Mean:                                    5154.7500
Sample Standard Deviation:                      8909.8733
Sample Coefficient of Variation:                   1.7285
 
Correlation:                                       0.9991
 
 
 
---------------------------------------------------------
  Confidence                         Lower          Upper
   Value (%)          Ratio          Limit          Limit
---------------------------------------------------------
      50.000         0.1603         0.1568         0.1676
      75.000         0.1603         0.1549         0.1892
      90.000         0.1603             **             **
      95.000         0.1603             **             **
      99.000         0.1603             **             **
      99.900         0.1603             **             **
      99.990         0.1603             **             **
      99.999         0.1603             **             **
    

 
. Step 1:   Define data
.
read y x
  0.1268825E+10  0.1246669E+10
  0.1295448E+10  0.1246669E+10
  0.1295448E+10  0.1268825E+10
  0.1168487E+08  0.1014325E+08
  0.1141398E+08  0.1014325E+08
  0.1168487E+08  0.1141398E+08
  0.3298360E+06  0.2902920E+06
  0.3298360E+06  0.1718490E+06
  0.2902920E+06  0.1718490E+06
  0.2415666E+07  0.1637297E+07
  0.2415666E+07  0.1347629E+07
  0.1637297E+07  0.1347629E+07
  0.9904356E+08  0.9530938E+08
  0.1049126E+09  0.9530938E+08
  0.1049126E+09  0.9904356E+08
  0.4930919E+08  0.4662120E+08
  0.4934958E+08  0.4662120E+08
  0.4934958E+08  0.4930919E+08
  0.1278483E+08  0.1232513E+08
  0.1286868E+08  0.1232513E+08
  0.1286868E+08  0.1278483E+08
  0.7029193E+07  0.4878485E+07
  0.7029193E+07  0.3244763E+07
  0.4878485E+07  0.3244763E+07
  0.1490000E+07  0.1040000E+07
  0.1860000E+07  0.1040000E+07
  0.1860000E+07  0.1490000E+07
  0.2680523E+07  0.2601516E+07
  0.2724237E+07  0.2601516E+07
  0.2724237E+07  0.2680523E+07
  0.8905137E+07  0.8303097E+07
  0.8905137E+07  0.8271071E+07
  0.8303097E+07  0.8271071E+07
  0.6956520E+06  0.6798450E+06
  0.6921780E+06  0.6798450E+06
  0.6956520E+06  0.6921780E+06
  0.3290000E+09  0.2890000E+09
  0.3300000E+09  0.2890000E+09
  0.3300000E+09  0.3290000E+09
  0.7091179E+05  0.6553055E+05
  0.7443393E+05  0.6553055E+05
  0.7443393E+05  0.7091179E+05
  0.8031739E+08  0.5416613E+08
  0.8031739E+08  0.4975062E+08
  0.5416613E+08  0.4975062E+08
  0.6830980E+07  0.6738330E+07
  0.6973430E+07  0.6738330E+07
  0.6973430E+07  0.6830980E+07
  0.2010000E+07  0.1980000E+07
  0.2600000E+07  0.2010000E+07
  0.2600000E+07  0.1980000E+07
  0.3193846E+08  0.3059341E+08
  0.3222820E+08  0.3059341E+08
  0.3222820E+08  0.3193846E+08
  0.1784258E+08  0.1460987E+08
  0.1784258E+08  0.1099276E+08
  0.1460987E+08  0.1099276E+08
  0.3150562E+09  0.3052555E+09
  0.3150562E+09  0.2994084E+09
  0.3052555E+09  0.2994084E+09
  0.7998000E+08  0.7574000E+08
  0.8017000E+08  0.7574000E+08
  0.8017000E+08  0.7998000E+08
  0.3983000E+08  0.3886000E+08
  0.4086000E+08  0.3886000E+08
  0.4086000E+08  0.3983000E+08
  0.2334030E+07  0.1387010E+07
  0.2544590E+07  0.2334030E+07
  0.2544590E+07  0.1387010E+07
  0.3126721E+09  0.2310785E+09
  0.2490103E+09  0.2310785E+09
  0.3126721E+09  0.2490103E+09
  0.1000900E+03  0.9977000E+02
  0.1255000E+03  0.9977000E+02
  0.1255000E+03  0.1000900E+03
  0.9860323E+04  0.9400626E+04
  0.9882525E+04  0.9400626E+04
  0.9882525E+04  0.9860323E+04
  0.2548997E+04  0.2482806E+04
  0.2640000E+08  0.2510000E+08
  0.2680000E+08  0.2510000E+08
  0.2680000E+08  0.2640000E+08
end of data
.
set write decimals 4
.
. Step 2:   Large sample interval
.
set ratio of means method large sample
ratio of means confidence limit y x
    

The following output is generated

            Confidence Limits for the Ratio of Means
              (Large Sample Approximation Method)
 
Numerator Variable:   Y
Denominator Variable: X
 
 
Summary Statistics for Numerator Variable:
Number of Observations:                                82
Sample Mean:                                98726120.5989
Sample Standard Deviation:                 251637033.8336
Sample Coefficient of Variation:                   2.5488
 
Summary Statistics for Variable 2:
Number of Observations:                                82
Sample Mean:                                92451883.4744
Sample Standard Deviation:                 243574789.7088
Sample Coefficient of Variation:                   2.6346
 
Correlation:                                       0.9988
 
 
 
---------------------------------------------------------
  Confidence                         Lower          Upper
   Value (%)          Ratio          Limit          Limit
---------------------------------------------------------
      50.000         1.0679         1.0557         1.0800
      75.000         1.0679         1.0470         1.0887
      90.000         1.0679         1.0380         1.0978
      95.000         1.0679         1.0321         1.1036
      99.000         1.0679         1.0205         1.1153
      99.900         1.0679         1.0065         1.1292
      99.990         1.0679         0.9943         1.1414
      99.999         1.0679         0.9832         1.1525
    

. set ratio of means method log ratio ratio of means confidence limit y x The following output is generated

            Confidence Limits for the Ratio of Means
                       (Log Ratio Method)
 
Numerator Variable:   Y
Denominator Variable: X
 
 
Summary Statistics for Numerator Variable:
Number of Observations:                                82
Sample Mean:                                98726120.5989
Sample Standard Deviation:                 251637033.8336
Sample Coefficient of Variation:                   2.5488
 
Summary Statistics for Variable 2:
Number of Observations:                                82
Sample Mean:                                92451883.4744
Sample Standard Deviation:                 243574789.7088
Sample Coefficient of Variation:                   2.6346
 
Correlation:                                       0.9988
 
 
 
---------------------------------------------------------
  Confidence                         Lower          Upper
   Value (%)          Ratio          Limit          Limit
---------------------------------------------------------
      50.000         1.0679         1.0558         1.0801
      75.000         1.0679         1.0472         1.0889
      90.000         1.0679         1.0384         1.0982
      95.000         1.0679         1.0327         1.1042
      99.000         1.0679         1.0215         1.1163
      99.900         1.0679         1.0082         1.1310
      99.990         1.0679         0.9968         1.1440
      99.999         1.0679         0.9865         1.1560
    

. set ratio of means method fieler ratio of means confidence limit y x The following output is generated

***** WARNING: RATIO OF MEANS CONFIDENCE LIMITS--
      FOR ALPHA (   99.99000    ), THE FIELLER INTERVAL IS UNBOUNDED.
 
***** WARNING: RATIO OF MEANS CONFIDENCE LIMITS--
      FOR ALPHA (   99.99900    ), THE FIELLER INTERVAL IS UNBOUNDED.
 
            Confidence Limits for the Ratio of Means
                        (Fieller Method)
 
Numerator Variable:   Y
Denominator Variable: X
 
 
Summary Statistics for Numerator Variable:
Number of Observations:                                82
Sample Mean:                                98726120.5989
Sample Standard Deviation:                 251637033.8336
Sample Coefficient of Variation:                   2.5488
 
Summary Statistics for Variable 2:
Number of Observations:                                82
Sample Mean:                                92451883.4744
Sample Standard Deviation:                 243574789.7088
Sample Coefficient of Variation:                   2.6346
 
Correlation:                                       0.9988
 
 
 
---------------------------------------------------------
  Confidence                         Lower          Upper
   Value (%)          Ratio          Limit          Limit
---------------------------------------------------------
      50.000         1.0679         1.0568         1.0818
      75.000         1.0679         1.0499         1.0951
      90.000         1.0679         1.0430         1.1148
      95.000         1.0679         1.0386         1.1334
      99.000         1.0679         1.0293         1.2096
      99.900         1.0679         1.0154         6.5994
      99.990         1.0679             **             **
      99.999         1.0679             **             **