NAG Library Function Document

nag_2_sample_t_test (g07cac)

1  Purpose

2  Specification

3  Description

1. It is assumed that the two variances are equal, that is ${\sigma }_x^2={\sigma }_y^2$.
   The test used is the two sample $t$-test. The test statistic $t$ is defined by;

   $$t_{\mathrm{obs}}=\frac{\stackrel{-}{x}-\stackrel{-}{y}}{s\sqrt{\left(1/n_x\right)+\left(1/n_y\right)}}$$

   where $s^2=\frac{\left(n_x-1\right)s_x^2+\left(n_y-1\right)s_y^2}{n_x+n_y-2}$ is the pooled variance of the two samples.
   Under the null hypothesis $H_0$ this test statistic has a $t$-distribution with $\left(n_x+n_y-2\right)$ degrees of freedom.
   The test of $H_0$ is carried out against one of three possible alternatives:

   1. $H_1:{\mu }_x\ne {\mu }_y$; the significance level, $p=P\left(t\ge \left|t_{\mathrm{obs}}\right|\right)$, i.e., a two tailed probability.
   2. $H_1:{\mu }_x>{\mu }_y$; the significance level, $p=P\left(t\ge t_{\mathrm{obs}}\right)$, i.e., an upper tail probability.
   3. $H_1:{\mu }_x<{\mu }_y$; the significance level, $p=P\left(t\le t_{\mathrm{obs}}\right)$, i.e., a lower tail probability.
   Upper and lower $100\left(1-\alpha \right)$% confidence limits for ${\mu }_x-{\mu }_y$ are calculated as:

   $$\left(\stackrel{-}{x}-\stackrel{-}{y}\right)\pm t_{1-\alpha /2}s\sqrt{\left(1/n_x\right)+\left(1/n_y\right)}\text{,}$$

   where $t_{1-\alpha /2}$ is the $100\left(1-\alpha /2\right)$ percentage point of the $t$-distribution with $\left(n_x+n_y-2\right)$ 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 $t^{\prime }$ is used where

   $$t_{\mathrm{obs}}^{\prime }=\frac{\stackrel{-}{x}-\stackrel{-}{y}}{\mathrm{se}\left(\stackrel{-}{x}-\stackrel{-}{y}\right)}$$

   where $\mathrm{se}\left(\stackrel{-}{x}-\stackrel{-}{y}\right)\left(\stackrel{-}{x}-\stackrel{-}{y}\right)=\sqrt{\frac{s_x^2}{n_x}+\frac{s_y^2}{n_y}}$.
   A $t$-distribution with $f$ degrees of freedom is used to approximate the distribution of $t^{\prime }$ where

   $$f=\frac{\mathrm{se}{\left(\stackrel{-}{x}-\stackrel{-}{y}\right)}^4}{\frac{s_x^2/n_x^2}{\left(n_x-1\right)}+\frac{s_y^2/n_y^2}{\left(n_y-1\right)}}\text{.}$$

   The test of $H_0$ is carried out against one of the three alternative hypotheses described above, replacing $t$ by $t^{\prime }$ and $t_{\mathrm{obs}}$ by $t_{\mathrm{obs}}^{\prime }$.
   Upper and lower $100\left(1-\alpha \right)$% confidence limits for ${\mu }_x-{\mu }_y$ are calculated as:

   $$\left(\stackrel{-}{x}-\stackrel{-}{y}\right)\pm t_{1-\alpha /2}\mathrm{se}\left(x-\stackrel{-}{y}\right)\text{.}$$

   where $t_{1-\alpha /2}$ is the $100\left(1-\alpha /2\right)$ percentage point of the $t$-distribution with $f$ degrees of freedom.

4  References

5  Arguments

1:     tail – Nag_TailProbabilityInput

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

$\mathbf{tail}=\mathrm{Nag\_TwoTail}$

The two tail probability, i.e., $H_1:{\mu }_x\ne {\mu }_y$.

$\mathbf{tail}=\mathrm{Nag\_UpperTail}$

The upper tail probability, i.e., $H_1:{\mu }_x>{\mu }_y$.

$\mathbf{tail}=\mathrm{Nag\_LowerTail}$

The lower tail probability, i.e., $H_1:{\mu }_x<{\mu }_y$.

Constraint: $\mathbf{tail}=\mathrm{Nag\_UpperTail}$, $\mathrm{Nag\_LowerTail}$ or $\mathrm{Nag\_TwoTail}$.

2:     equal – Nag_PopVarInput

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

$\mathbf{equal}=\mathrm{Nag\_PopVarEqual}$

The population variances are assumed to be equal, that is ${\sigma }_x^2={\sigma }_y^2$.

$\mathbf{equal}=\mathrm{Nag\_PopVarNotEqual}$

The population variances are not assumed to be equal.

Constraint: $\mathbf{equal}=\mathrm{Nag\_PopVarEqual}$ or $\mathrm{Nag\_PopVarNotEqual}$.

3:     nx – IntegerInput

On entry: the size of the $X$ sample, $n_x$.

Constraint: $\mathbf{nx}\ge 2$.

4:     ny – IntegerInput

On entry: the size of the $Y$ sample, $n_y$.

Constraint: $\mathbf{ny}\ge 2$.

5:     xmean – doubleInput

On entry: the mean of the $X$ sample, $\stackrel{-}{x}$.

6:     ymean – doubleInput

On entry: the mean of the $Y$ sample, $\stackrel{-}{y}$.

7:     xstd – doubleInput

On entry: the standard deviation of the $X$ sample, $s_x$.

Constraint: $\mathbf{xstd}>0.0$.

8:     ystd – doubleInput

On entry: the standard deviation of the $Y$ sample, $s_y$.

Constraint: $\mathbf{ystd}>0.0$.

9:     clevel – doubleInput

On entry: the confidence level, $1-\alpha$, for the specified tail. For example $\mathbf{clevel}=0.95$ will give a 95% confidence interval.

Constraint: $0.0<\mathbf{clevel}<1.0$.

10:   t – double *Output

On exit: contains the test statistic, $t_{\mathrm{obs}}$ or $t_{\mathrm{obs}}^{\prime }$.

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 ${\mu }_x-{\mu }_y$.

14:   du – double *Output

On exit: contains the upper confidence limit for ${\mu }_x-{\mu }_y$.

15:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

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, $\mathbf{nx}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nx}\ge 2$.

On entry, $\mathbf{ny}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ny}\ge 2$.

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: $\mathbf{clevel}=〈\mathit{\text{value}}〉$.

NE_REAL_ARG_LE

On entry, clevel must not be less than or equal to 0.0: $\mathbf{clevel}=〈\mathit{\text{value}}〉$.

On entry, xstd must not be less than or equal to 0.0: $\mathbf{xstd}=〈\mathit{\text{value}}〉$.

On entry, ystd must not be less than or equal to 0.0: $\mathbf{ystd}=〈\mathit{\text{value}}〉$.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (g07cace.c)

9.2  Program Data

Program Data (g07cace.d)

9.3  Program Results

Program Results (g07cace.r)