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

2 Specification

#include <nag.h>

void	g07cac (Nag_TailProbability tail, Nag_PopVar equal, Integer nx, Integer ny, double xmean, double ymean, double xstd, double ystd, double clevel, double t, double df, double prob, double dl, double du, NagError fail)

The function may be called by the names: g07cac, nag_univar_ttest_2normal or nag_2_sample_t_test.

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.

g07cac 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 σ x 2 = σ y 2 .
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:
1. (i) $H_{1} : μ_{x} \neq μ_{y}$ ; the significance level, $p = P (t \geq | t_{obs} |)$ , i.e., a two tailed probability.
2. (ii) $H_{1} : μ_{x} > μ_{y}$ ; the significance level, $p = P (t \geq t_{obs})$ , i.e., an upper tail probability.
3. (iii) $H_{1} : μ_{x} < μ_{y}$ ; 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 (1979). 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}) (\bar{x} - \bar{y}) = \sqrt{\frac{s_{x}^{2}}{n_{x}} + \frac{s_{y}^{2}}{n_{y}}}

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 \times t^{'}

and

t_{obs} \times 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.

4 References

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

Kendall M G and Stuart A (1979) The Advanced Theory of Statistics (3 Volumes) (4th Edition) Griffin

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

5 Arguments

1: $tail$ – Nag_TailProbability Input

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

$tail = Nag_TwoTail$: The two tail probability, i.e., $H_{1} : μ_{x} \neq μ_{y}$ .
$tail = Nag_UpperTail$: The upper tail probability, i.e., $H_{1} : μ_{x} > μ_{y}$ .
$tail = Nag_LowerTail$: The lower tail probability, i.e., $H_{1} : μ_{x} < μ_{y}$ .

Constraint:

tail = Nag_UpperTail

Nag_LowerTail

Nag_TwoTail

2: $equal$ – Nag_PopVar Input

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

$equal = Nag_PopVarEqual$: The population variances are assumed to be equal, that is $σ_{x}^{2} = σ_{y}^{2}$ .
$equal = Nag_PopVarNotEqual$: The population variances are not assumed to be equal.

Constraint:

equal = Nag_PopVarEqual

Nag_PopVarNotEqual

3: $nx$ – Integer Input

On entry: the size of the

X

sample,

n_{x}

Constraint:

nx \geq 2

4: $ny$ – Integer Input

On entry: the size of the

Y

sample,

n_{y}

Constraint:

ny \geq 2

5: $xmean$ – double Input

On entry: the mean of the

X

sample,

\bar{x}

6: $ymean$ – double Input

On entry: the mean of the

Y

sample,

\bar{y}

7: $xstd$ – double Input

On entry: the standard deviation of the

X

sample,

s_{x}

Constraint:

xstd > 0.0

8: $ystd$ – double Input

On entry: the standard deviation of the

Y

sample,

s_{y}

Constraint:

ystd > 0.0

9: $clevel$ – double 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: $t$ – double * Output

On exit: contains the test statistic,

t_{obs}

t_{obs}^{'}

11: $df$ – double * Output

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

12: $prob$ – double * Output

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

p

, as defined by tail.

13: $dl$ – double * Output

On exit: contains the lower confidence limit for

μ_{x} - μ_{y}

14: $du$ – double * Output

On exit: contains the upper confidence limit for

μ_{x} - μ_{y}

15: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument equal had an illegal value.

On entry, argument tail had an illegal value.
NE_INT_ARG_LT: On entry, $nx = ⟨ value ⟩$ .
Constraint: $nx \geq 2$ .

On entry, $ny = ⟨ value ⟩$ .
Constraint: $ny \geq 2$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL_ARG_GE: On entry, clevel must not be greater than or equal to 1.0: $clevel = ⟨ value ⟩$ .
NE_REAL_ARG_LE: On entry, clevel must not be less than or equal to 0.0: $clevel = ⟨ value ⟩$ .

On entry, xstd must not be less than or equal to 0.0: $xstd = ⟨ value ⟩$ .

On entry, ystd must not be less than or equal to 0.0: $ystd = ⟨ value ⟩$ .

7 Accuracy

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

8 Parallelism and Performance

g07cac is not threaded in any implementation.

9 Further Comments

The sample means and standard deviations can be computed using g01atc.

10 Example

The following example program 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.

g07ca: FL CL CPP AD

NAG CL Interfaceg07cac (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 CL Interface
g07cac (ttest_2normal)