G07CAF : NAG Library, Mark 24

3 Description

Consider two independent samples, denoted by

X

and

Y

, of size

n_{x}

and

n_{y}

drawn from two Normal populations with means

μ_{x}

and

μ_{y}

, and variances

σ_{x}^{2}

and

σ_{y}^{2}

respectively. Denote the sample means by

\bar{x}

and

\bar{y}

and the sample variances by

s_{x}^{2}

and

s_{y}^{2}

respectively.

G07CAF calculates a test statistic and its significance level to test the null hypothesis

H_{0} : μ_{x} = μ_{y}

, together with upper and lower confidence limits for

μ_{x} - μ_{y}

. The test used depends on whether or not the two population variances are assumed to be equal.

It is assumed that the two variances are equal, that is σx2=σy2.
The test used is the two sample $t$ -test. The test statistic $t$ is defined by;
$t_{obs} = \frac{\bar{x} - \bar{y}}{s \sqrt{(1 / n_{x}) + (1 / n_{y})}}$
where
$s^{2} = \frac{(n_{x} - 1) s_{x}^{2} + (n_{y} - 1) s_{y}^{2}}{n_{x} + n_{y} - 2}$
is the pooled variance of the two samples.

Under the null hypothesis $H_{0}$ this test statistic has a $t$ -distribution with $(n_{x} + n_{y} - 2)$ degrees of freedom.
The test of H0 is carried out against one of three possible alternatives;
- $H_{1} : μ_{x} \neq μ_{y}$ ; the significance level, $p = P (t \geq |t_{obs}|)$ , i.e., a two tailed probability.
- $H_{1} : μ_{x} > μ_{y}$ ; the significance level, $p = P (t \geq t_{obs})$ , i.e., an upper tail probability.
- $H_{1} : μ_{x} < μ_{y}$ ; the significance level, $p = P (t \leq t_{obs})$ , i.e., a lower tail probability.
Upper and lower $100 (1 - α) %$ confidence limits for $μ_{x} - μ_{y}$ are calculated as:
$(\bar{x} - \bar{y}) \pm t_{1 - α / 2} s \sqrt{(1 / n_{x}) + (1 / n_{y})} .$
where $t_{1 - α / 2}$ is the $100 (1 - α / 2)$ percentage point of the $t$ -distribution with ( $n_{x} + n_{y} - 2$ ) degrees of freedom.
It is not assumed that the two variances are equal.
If the population variances are not equal the usual two sample $t$ -statistic no longer has a $t$ -distribution and an approximate test is used.

This problem is often referred to as the Behrens–Fisher problem, see Kendall and Stuart (1969). The test used here is based on Satterthwaites procedure. To test the null hypothesis the test statistic $t^{'}$ is used where
$t_{obs}^{'} = \frac{\bar{x} - \bar{y}}{se (\bar{x} - \bar{y})}$
where $se (\bar{x} - \bar{y}) = \sqrt{\frac{s_{x}^{2}}{n_{x}} + \frac{s_{y}^{2}}{n_{y}}}$ .

A $t$ -distribution with $f$ degrees of freedom is used to approximate the distribution of $t^{'}$ where
$f = \frac{se {(\bar{x} - \bar{y})}^{4}}{\frac{{(s_{x}^{2} / n_{x})}^{2}}{(n_{x} - 1)} + \frac{{(s_{y}^{2} / n_{y})}^{2}}{(n_{y} - 1)}} .$
The test of $H_{0}$ is carried out against one of the three alternative hypotheses described above, replacing $t$ by $t^{'}$ and $t_{obs}$ by $t_{obs}^{'}$ .

Upper and lower $100 (1 - α) %$ confidence limits for $μ_{x} - μ_{y}$ are calculated as:
$(\bar{x} - \bar{y}) \pm t_{1 - α / 2} se (x - \bar{y}) .$
where $t_{1 - α / 2}$ is the $100 (1 - α / 2)$ percentage point of the $t$ -distribution with $f$ degrees of freedom.

5 Parameters

1: TAIL – CHARACTER(1)Input

On entry: indicates which tail probability is to be calculated, and thus which alternative hypothesis is to be used.

$TAIL ='T'$: The two tail probability, i.e., $H_{1} : μ_{x} \neq μ_{y}$ .
$TAIL ='U'$: The upper tail probability, i.e., $H_{1} : μ_{x} > μ_{y}$ .
$TAIL ='L'$: The lower tail probability, i.e., $H_{1} : μ_{x} < μ_{y}$ .

Constraint:

TAIL ='T'

'U'

'L'

2: EQUAL – CHARACTER(1)Input

On entry: indicates whether the population variances are assumed to be equal or not.

$EQUAL ='E'$: The population variances are assumed to be equal, that is $σ_{x}^{2} = σ_{y}^{2}$ .
$EQUAL ='U'$: The population variances are not assumed to be equal.

Constraint:

EQUAL ='E'

'U'

3: NX – INTEGERInput

On entry:

n_{x}

, the size of the

X

sample.

Constraint:

NX \geq 2

4: NY – INTEGERInput

On entry:

n_{y}

, the size of the

Y

sample.

Constraint:

NY \geq 2

5: XMEAN – REAL (KIND=nag_wp)Input

On entry:

\bar{x}

, the mean of the

X

sample.

6: YMEAN – REAL (KIND=nag_wp)Input

On entry:

\bar{y}

, the mean of the

Y

sample.

7: XSTD – REAL (KIND=nag_wp)Input

On entry:

s_{x}

, the standard deviation of the

X

sample.

Constraint:

XSTD > 0.0

8: YSTD – REAL (KIND=nag_wp)Input

On entry:

s_{y}

, the standard deviation of the

Y

sample.

Constraint:

YSTD > 0.0

9: CLEVEL – REAL (KIND=nag_wp)Input

On entry: the confidence level,

1 - α

, for the specified tail. For example

CLEVEL = 0.95

will give a

95 %

confidence interval.

Constraint:

0.0 < CLEVEL < 1.0

10: T – REAL (KIND=nag_wp)Output

On exit: contains the test statistic,

t_{obs}

t_{obs}^{'}

11: DF – REAL (KIND=nag_wp)Output

On exit: contains the degrees of freedom for the test statistic.

12: PROB – REAL (KIND=nag_wp)Output

On exit: contains the significance level, that is the tail probability,

p

, as defined by TAIL.

13: DL – REAL (KIND=nag_wp)Output

On exit: contains the lower confidence limit for

μ_{x} - μ_{y}

14: DU – REAL (KIND=nag_wp)Output

On exit: contains the upper confidence limit for

μ_{x} - μ_{y}

15: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

IFAIL = 1

On entry,	$TAIL \neq'T'$ , $'U'$ or $'L'$ ,
or	$EQUAL \neq'E'$ or $'U'$ ,
or	$NX < 2$ ,
or	$NY < 2$ ,
or	$XSTD \leq 0.0$ ,
or	$YSTD \leq 0.0$ ,
or	$CLEVEL \leq 0.0$ ,
or	$CLEVEL \geq 1.0$ .

9 Example

This example reads the two sample sizes and the sample means and standard deviations for two independent samples. The data is taken from page 116 of Snedecor and Cochran (1967) from a test to compare two methods of estimating the concentration of a chemical in a vat. A test of the equality of the means is carried out first assuming that the two population variances are equal and then making no assumption about the equality of the population variances.

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G07CAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG07CAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G07CAF