g07caf (ttest_2normal) : NAG Library, Mark 27

g07caf computes a

t

-test statistic to test for a difference in means between two Normal populations, together with a confidence interval for the difference between the means.

The routine may be called by the names g07caf or nagf_univar_ttest_2normal.

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.

1.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 $H_{0}$ 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.
2.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.

Johnson M G and Kotz A (1969) The Encyclopedia of Statistics 2 Griffin

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

Snedecor G W and Cochran W G (1967) Statistical Methods Iowa State University Press

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'

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'

On entry:

n_{x}

, the size of the

X

sample.

Constraint:

nx \geq 2

On entry:

n_{y}

, the size of the

Y

sample.

Constraint:

ny \geq 2

On entry:

\bar{x}

, the mean of the

X

sample.

On entry:

\bar{y}

, the mean of the

Y

sample.

On entry:

s_{x}

, the standard deviation of the

X

sample.

Constraint:

xstd > 0.0

On entry:

s_{y}

, the standard deviation of the

Y

sample.

Constraint:

ystd > 0.0

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

On exit: contains the test statistic,

t_{obs}

t_{obs}^{'}

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

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

p

, as defined by tail.

On exit: contains the lower confidence limit for

μ_{x} - μ_{y}

On exit: contains the upper confidence limit for

μ_{x} - μ_{y}

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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).

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:

On entry,

clevel = ⟨ value ⟩

.
Constraint:

0.0 < clevel < 1.0

On entry,

equal = ⟨ value ⟩

.
Constraint:

equal ='E'

'U'

On entry,

nx = ⟨ value ⟩

.
Constraint:

nx \geq 2

On entry,

ny = ⟨ value ⟩

.
Constraint:

ny \geq 2

On entry,

tail = ⟨ value ⟩

.
Constraint:

tail ='T'

'U'

'L'

On entry,

xstd = ⟨ value ⟩

.
Constraint:

xstd > 0.0

On entry,

ystd = ⟨ value ⟩

.
Constraint:

ystd > 0.0

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

Your licence key may have expired or may not have been installed correctly.

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

The computed probability and the confidence limits should be accurate to approximately five significant figures.

g07caf is not threaded in any implementation.

The sample means and standard deviations can be computed using g01atf.

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.

NAG FL Interface
g07caf (ttest_2normal)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interfaceg07caf (ttest_​2normal)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g07caf (ttest_2normal)