ttest#

Purpose#

Performs two-sample and paired t-tests for comparing means between groups.

Format#

out = ttest(y1, y2[, ctl])#

out = ttest(data, formula[, ctl])

out = ttest(filename, formula[, ctl])

Parameters:

y1 (Nx1 vector) – first sample.
y2 (Mx1 vector) – second sample.
data (dataframe) – dataframe containing variables.
filename (string) – name of dataset.
formula (string) – formula string of the form "y ~ group".

ctl (struct) –

Optional argument, instance of a ttestControl structure containing the following members:

ctl.output	scalar, print results. Default = 1. 1: Print results. 0: Suppress output.
ctl.paired	scalar, test type. Default = 0. 0: Independent samples t-test. 1: Paired samples t-test.
ctl.alternative	scalar, alternative hypothesis. Default = 0. 0: Two-sided (mean1 != mean2). 1: Greater (mean1 > mean2). -1: Less (mean1 < mean2).
ctl.mu	scalar, null hypothesis difference in means. Default = 0.
ctl.varEqual	scalar, variance assumption. Default = 0. 0: Welch t-test (unequal variances). 1: Pooled variance (equal variances assumed).
ctl.confLevel	scalar, confidence level for confidence interval. Default = 0.95.
ctl.miss	scalar, missing value handling. Default = 0. 0: Error if missing values present. 1: Listwise deletion of missing values.

Returns:

out (struct) –

instance of ttestOut structure:

out.groups	2x1 string array, group labels.
out.mean	2x1 vector, group means.
out.sd	2x1 vector, group standard deviations.
out.n	2x1 vector, group sample sizes.
out.meanDiff	scalar, difference in means (group1 - group2).
out.tEq	scalar, t-statistic assuming equal variances.
out.dfEq	scalar, degrees of freedom (equal variances).
out.pEq	scalar, p-value (equal variances).
out.tWelch	scalar, t-statistic (Welch).
out.dfWelch	scalar, degrees of freedom (Satterthwaite approximation).
out.pWelch	scalar, p-value (Welch).
out.fStat	scalar, F-statistic for variance equality test.
out.dfF	2x1 vector, numerator and denominator df for F-test.
out.pF	scalar, p-value for F-test of equal variances.
out.ci	1x2 vector, confidence interval for mean difference.
out.confLevel	scalar, confidence level used.

Examples#

Example 1: Two-sample t-test with vectors#

// Two independent samples
y1 = { 23, 25, 28, 22, 24 };
y2 = { 30, 32, 29, 35, 31 };

// Perform t-test
out = ttest(y1, y2);

Two-Sample t-test

Descriptive Statistics
------------------------------------------------------------
          Group         N        Mean     Std Dev
        Group 1         5     24.4000      2.3022
        Group 2         5     31.4000      2.3022

Mean Difference:      -7.0000

Test of Means
------------------------------------------------------------
              Method         t        df     p-value
 Welch (unequal var)   -4.8076       8.0      0.0013
  Pooled (equal var)   -4.8076         8      0.0013

Test of Variances
------------------------------------------------------------
                             F        df     p-value
              F-test    1.0000     4,  4      1.0000

 95% Confidence Interval: [  -10.3576,    -3.6424]

The small p-value (0.0013) indicates a statistically significant difference between the two groups.

Example 2: Paired t-test#

// Before and after measurements
before = { 120, 125, 118, 130, 122 };
after = { 115, 120, 112, 125, 118 };

struct ttestControl ctl;
ctl = ttestControlCreate();
ctl.paired = 1;

out = ttest(before, after, ctl);

Paired Samples t-test

Descriptive Statistics
------------------------------------------------------------
          Group         N        Mean     Std Dev
        Group 1         5    123.0000      4.6904
        Group 2         5    118.0000      4.9497

Mean Difference:       5.0000

Paired t-test
------------------------------------------------------------
                             t        df     p-value
              Paired   15.8114         4      0.0001

 95% Confidence Interval: [    4.1220,     5.8780]

Example 3: One-sided test#

y1 = { 23, 25, 28, 22, 24 };
y2 = { 30, 32, 29, 35, 31 };

struct ttestControl ctl;
ctl = ttestControlCreate();
ctl.alternative = 1;  // test if group1 > group2

out = ttest(y1, y2, ctl);

The one-sided p-value (0.9993) is large because group 1’s mean (24.4) is actually less than group 2’s mean (31.4), contradicting the alternative hypothesis that group 1 > group 2.

Example 4: Using dataframe with formula#

// Load data
data = loadd(getGAUSSHome("examples/auto2.dta"), "mpg + foreign");

// Test if mpg differs by origin
struct ttestControl ctl;
ctl = ttestControlCreate();

out = ttest(data, "mpg ~ foreign", ctl);

Two-Sample t-test

Descriptive Statistics
------------------------------------------------------------
          Group         N        Mean     Std Dev
        Group 1        52     19.8269      4.7433
        Group 2        22     24.7727      6.6112

Mean Difference:      -4.9458

Test of Means
------------------------------------------------------------
              Method         t        df     p-value
 Welch (unequal var)   -3.1797      30.5      0.0034
  Pooled (equal var)   -3.6308        72      0.0005

Test of Variances
------------------------------------------------------------
                             F        df     p-value
              F-test    1.9427    21, 51      0.0549

 95% Confidence Interval: [   -8.1201,    -1.7716]

Foreign cars average 4.9 more mpg than domestic. The Welch test (p = 0.0034) confirms the difference is statistically significant.

Remarks#

By default, performs Welch’s t-test which does not assume equal variances. Set ctl.varEqual = 1 for the pooled variance version.
The output includes both equal-variance and Welch statistics for comparison.
An F-test for equality of variances is automatically computed.
For paired tests, both samples must have the same length.