naginterfaces.library.univar.ttest_​2normal

naginterfaces.library.univar.ttest_2normal(tail, equal, nx, ny, xmean, ymean, xstd, ystd, clevel)

ttest_2normal 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.

For full information please refer to the NAG Library document for g07ca

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g07/g07caf.html

Parameters

tailstr, length 1

Indicates which tail probability is to be calculated, and thus which alternative hypothesis is to be used.

$tail=\text{'T'}$

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

$tail=\text{'U'}$

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

$tail=\text{'L'}$

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

equalstr, length 1

Indicates whether the population variances are assumed to be equal or not.

$equal=\text{'E'}$

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

$equal=\text{'U'}$

The population variances are not assumed to be equal.

nxint

$n_x$, the size of the $X$ sample.

nyint

$n_y$, the size of the $Y$ sample.

xmeanfloat

$\overline{x}$, the mean of the $X$ sample.

ymeanfloat

$\overline{y}$, the mean of the $Y$ sample.

xstdfloat

$s_x$, the standard deviation of the $X$ sample.

ystdfloat

$s_y$, the standard deviation of the $Y$ sample.

clevelfloat

The confidence level, $1-\alpha$, for the specified tail. For example $clevel=0.95$ will give a $95\%$ confidence interval.

Returns

tfloat

Contains the test statistic, $t_{obs}$ or $t_{obs}^{\prime }$.

dffloat

Contains the degrees of freedom for the test statistic.

probfloat

Contains the significance level, that is the tail probability, $p$, as defined by $tail$.

dlfloat

Contains the lower confidence limit for ${\mu }_x-{\mu }_y$.

dufloat

Contains the upper confidence limit for ${\mu }_x-{\mu }_y$.

Raises

NagValueError

(errno $1$)

On entry, $clevel=⟨value⟩$.

Constraint: $0.0<clevel<1.0$.

(errno $1$)

On entry, $ystd=⟨value⟩$.

Constraint: $ystd>0.0$.

(errno $1$)

On entry, $xstd=⟨value⟩$.

Constraint: $xstd>0.0$.

(errno $1$)

On entry, $ny=⟨value⟩$.

Constraint: $ny\ge 2$.

(errno $1$)

On entry, $nx=⟨value⟩$.

Constraint: $nx\ge 2$.

(errno $1$)

On entry, $equal=⟨value⟩$.

Constraint: $equal=\text{'E'}$ or $\text{'U'}$.

(errno $1$)

On entry, $tail=⟨value⟩$.

Constraint: $tail=\text{'T'}$, $\text{'U'}$ or $\text{'L'}$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

Consider two independent samples, denoted by $X$ and $Y$, of size $n_x$ and $n_y$ drawn from two Normal populations with means ${\mu }_x$ and ${\mu }_y$, and variances ${\sigma }_x^2$ and ${\sigma }_y^2$ respectively. Denote the sample means by $\overline{x}$ and $\overline{y}$ and the sample variances by $s_x^2$ and $s_y^2$ respectively.

ttest_2normal calculates a test statistic and its significance level to test the null hypothesis $H_0:{\mu }_x={\mu }_y$, together with upper and lower confidence limits for ${\mu }_x-{\mu }_y$. 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 ${\sigma }_x^2={\sigma }_y^2$.

   The test used is the two sample $t$-test. The test statistic $t$ is defined by;

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

   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;

   $H_1:{\mu }_x\ne {\mu }_y$; the significance level, $p=P(t\ge \left|t_{obs}\right|)$, i.e., a two tailed probability.

   $H_1:{\mu }_x>{\mu }_y$; the significance level, $p=P(t\ge t_{obs})$, i.e., an upper tail probability.

   $H_1:{\mu }_x<{\mu }_y$; the significance level, $p=P(t\le t_{obs})$, i.e., a lower tail probability.

   Upper and lower $100(1-\alpha )\%$ confidence limits for ${\mu }_x-{\mu }_y$ are calculated as:

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

   where $t_{1-\alpha /2}$ is the $100(1-\alpha /2)$ 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 $t^{\prime }$ is used where

   $$t_{obs}^{\prime }=\frac{\overline{x}-\overline{y}}{se(\overline{x}-\overline{y})}$$

   where $se(\overline{x}-\overline{y})=\sqrt{\frac{s_x^2}{n_x}+\frac{s_y^2}{n_y}}$.

   A $t$-distribution with $f$ degrees of freedom is used to approximate the distribution of $t^{\prime }$ where

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

   The test of $H_0$ is carried out against one of the three alternative hypotheses described above, replacing $t$ by $t^{\prime }$ and $t_{obs}$ by $t_{obs}^{\prime }$.

   Upper and lower $100(1-\alpha )\%$ confidence limits for ${\mu }_x-{\mu }_y$ are calculated as:

   $$(\overline{x}-\overline{y})\pm t_{1-\alpha /2}se(x-\overline{y})\text{.}$$

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

References

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