Difference Tests: R, %, Means

Select Difference tests: r, %, means on the Basic Statistics and Tables Startup Panel - Quick tab to display the Difference tests: r, %, means dialog box. These options are used to compute a variety of significance tests including the difference between two correlation coefficients, the difference between two proportions, and the difference between two means when the standard deviation is known.

Send/print results for each Compute to Report window. Select the Send/print results for each Compute to Report window check box if you want to automatically send the results of the selected probability computations to a report when you click a Compute button. This option is available only if you have specified in the Output Manager that the output is to be sent to a report window.

Difference between two correlation coefficients. The options in this group box are used to compare the statistical significance of differences between two Pearson r's. See also Correlations.

r1. Enter the Pearson r (correlation coefficient) of the first sample.

r2. Enter the Pearson r of the second sample.

N1. Enter the sample size (number of cases) of the first sample.

N2. Enter the sample size of the second sample.

Compute.  After you have entered the values, click the Compute button to calculate the p-value. Both One-sided and Two-sided tests can be performed.

Note: The difference between two correlation coefficients is computed using the r-to-Fisher-z transformation:

where:

r' - is the Fisher-z transformed (to a normally distributed variate) Pearson correlation coefficient r

r - is the standard Pearson correlation coefficient

Given the parameters, as for example, in the following dialog image:

The significance of the difference between two correlation coefficients is computed as follows:

where d - is the difference between the two Fisher z-transformed correlation coefficients

where:

sd - is the standard error of the difference between the two normalized (Fisher-z transformed) correlation coefficients

n1, n2 - are the two sample sizes (for r1 and r2, respectively)

The test statistic d/sd is then evaluated against the t distribution with df = n1 + n2 -4 degrees of freedom. The one-sided and two-sided p values are computed as usual, by considering either both sides or only one side of the t distribution.

Difference between two means (normal distribution). These options enable you to compute:

1. The significance level for the difference between two means computed from two samples;

2. The significance level for the difference between one mean computed from a sample and a population mean (if the Single mean 1 vs. population mean 2 check box is selected).

M 1. Enter the mean of the first sample.

M 2. Enter the mean of the second sample.

StDv 1. Enter the standard deviation of the first sample.

StDv 2. Enter the standard deviation of the second sample.

N1. Enter the sample size (number of samples) of the first sample.

N2. Enter the sample size of the second sample.

Compute.  After you have entered the values, click the Compute button to calculate the p-value. Both One-sided and Two-sided tests can be performed. The p-value is computed based on the t-value for the respective comparison. See also t-test for independent samples.

Note: Difference between two sample means:

When two means are to be compared, the difference between the two means is evaluated as follows.

where d - is the difference between the two means m1 and m2

where:

sd - is the standard error of the difference between means, computed from the pooled variance estimates for the two means (s12 and s22)

s12, s22 - are the variance estimates (standard deviations squared) for the two means

n1, n2  - are the two sample sizes (for the two means m1 and m2, respectively)

The test statistic d/sd is then evaluated against the t distribution with df = n1 + n2 -2 degrees of freedom. The one-sided and two-sided p values are computed as usual, by considering either both sides or only one side of the t distribution.

Note: Difference between a sample mean and a population mean:

When the Single mean 1 versus population mean 2 check box is selected, the program will evaluate the statistical significance of the difference as follows.

where d - is the difference between the sample mean m1 and the population mean µ2

where:

sd - is the standard error of the difference between mean m1

s1 - is the standard deviation the sample (from which m1 was computed)

n1 - is the sample size

The test statistic d/sd is then evaluated against the t distribution with df = n1 - 1 degrees of freedom. The one-sided and two-sided p values are computed as usual, by considering either both sides or only one side of the t distribution.

Difference between two proportions. These options are used to compute the significance level for the difference between two proportions.

Pr. 1. Enter the proportion of the first sample.

Pr. 2. Enter the proportion of the second sample.

N1. Enter the sample size (number of samples) of the first sample.

N2. Enter the sample size of the second sample.

Compute.  After you have entered the values, click the Compute button to calculate the p-value. Both One-sided and Two-sided tests can be performed.

Note: The p-value is computed based on the z-value for the respective comparison:

|z|=√[(N1*N2)/(N1+N2)]*|p1-p2|/√(p*q)

where

p=(p1*N1+p2* N2)/(N1+N2)

q=1-p.