Basic classical statistical tests for 2 correlated samples (stratified)

Basic classical statistical tests for 2 correlated samples (stratified)#

Student t-test for 2 correlated samples: tests (p-values)#

ctx.studentt_2csamples_test(mean, sd, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns results for Student’s t-test for for 2 correlated samples.

Parameters:

Mean:: The mean of the sample.
Sd:: The standard deviation of the sample
N:: The sample size
Alpha:: The alpha-level used for confidence intervals

Let \((X_1, X_2, \ldots, X_N)\) denote a random sample of size \(N\) from a normal distribution with mean \(\mu\) and variance \(\sigma^2\), and let

\[\overline{x}_1 = \frac{1}{N} \sum_{i=1}^N X_i \quad \text{and } s^2 = \frac{1}{N-1} \sum_{i=1}^N (X_i - \overline{x}_1)\]

be the usual sample estimates of the unkown population mean \(\mu\) and unkown population variance \(\sigma^2\). Then Student’s t-test can be used to test hypotheses concerning \(\mu\) with regard to a reference value \(\mu_2\).

Let \(F_t\left(\cdot, \nu\right)\) denote the CDF (see section ref{tDistributionCDF}) and let \(t_{\nu,\alpha}\) denote the \(\alpha\)-quantile (see section ref{tDistributionQuantile}) of the \(t\)-distribution with \(\nu\) degrees of freedom. Define

\[t= \frac{\overline{x}_1-\mu_2}{s}, \quad s=\sqrt{s_1^2 /N}, \quad \nu=N-1.\]

Then \(p\)-values and rejection criteria for \(H_0\) can be calculated as summarized below

Test problem

\(p\)-value

Reject \(H_0\)

\(H_{01}: \mu_1\leq \mu_2\) vs \(H_{A1}: \mu_1> \mu_2\)

\(F_t\left(-t, \nu\right)\)

\(t > t_{\nu;1-\alpha}\)

\(H_{02}: \mu_1\geq \mu_2\) vs \(H_{A2}: \mu_1< \mu_2\)

\(F_t\left(t, \nu\right)\)

\(t > t_{\nu;\alpha}\)

\(H_{03}: \mu_1= \mu_2\) vs \(H_{A3}: \mu_1\neq \mu_2\)

\(F_t\left(t, \nu\right)-F_t\left(-t, \nu\right)\)

\(t > t_{\nu;1-\alpha/2}\) or \(t > t_{\nu;\alpha/2}\)

Test problem	\(p\)-value	Reject \(H_0\)
\(H_{01}: \mu_1\leq \mu_2\) vs \(H_{A1}: \mu_1> \mu_2\)	\(F_t\left(-t, \nu\right)\)	\(t > t_{\nu;1-\alpha}\)
\(H_{02}: \mu_1\geq \mu_2\) vs \(H_{A2}: \mu_1< \mu_2\)	\(F_t\left(t, \nu\right)\)	\(t > t_{\nu;\alpha}\)
\(H_{03}: \mu_1= \mu_2\) vs \(H_{A3}: \mu_1\neq \mu_2\)	\(F_t\left(t, \nu\right)-F_t\left(-t, \nu\right)\)	\(t > t_{\nu;1-\alpha/2}\) or \(t > t_{\nu;\alpha/2}\)

The test can also be expressed in terms of a correlation coefficient \(r\) between the combined \(X\) and an indicator variable, where \(t\) and \(r\) are related by

\[r=\frac{t}{\sqrt{t^2+\nu}}, \quad t= \nu \frac{r}{1-r^2}.\]

An actual call to the function, requesting Student’s t-test with description, the critical value for a two-sided test, the p-value for \(H_{03}\) (in the case of \(\textsf{TTest}\) this is \(\mu_1 \neq \mu_2\)), for 2 independent samples of size 10 and standard deviation 1 each, with means 2.3 and 4.5, and a type I error \(\alpha=0.05\) would be
>>> from mpfunlab import mpm
>>> xreal.StudentT1Test(means:=[5.24, 4.05], sd:=1.5, n:=22)

Input
Variable Variable 1
Common N 22
Mean. Group 1 5.24
Mean. Group 2 4.05
StDev. Group 1 1.5
StDev. Group 2 3.5
Pearson's rho 0.7
Pearson's rho0. reference 0.2

Student's t-test for 2 corr. samples
Parameter Result
df 21
Difference of means 1.19
t-value (=delta) 2.087398086
t1 - alpha(1 - sided) 1.720742903
t1 - alpha(2 - sided) 2.079613845
test. p-value (H01: µ1 >= µ2) 0.97538843
test. p-value (H02: µ1 <= µ2) 0.02461157
test. p-value (H03: µ1 = µ2) 0.04922314

Student t-test for 2 correlated samples: confidence intervals#

ctx.studentt_2csamples_ci(mean, sd, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns results for Student’s t-test for for 2 correlated samples.

Parameters:

Mean:: The mean of the sample.
Sd:: The standard deviation of the sample
N:: The sample size
Alpha:: The alpha-level used for confidence intervals

Let \(A_1=t_{\nu,\alpha} \cdot s\) and \(A_2=t_{\nu,\alpha/2} \cdot s\), where \(s\) and \(\nu\) are defined in (ref{eq:TTest1}), and \(t_{\nu,\alpha}\) denotes the \(\alpha\)-quantile of the (central) \(t\)-distribution with \(\nu\) degrees of freedom (see section ref{tDistributionQuantile}).

Type

Confidence Interval (Difference of Means)

Left-sided

\(-\infty \leq \mu_1 - \mu_2 \leq (\overline{x}_1-\mu_2) + A_1\)

Right-sided

\((\overline{x}_1-\mu_2 ) - A_1 \leq \mu_1 - \mu_2 \leq +\infty\)

Two-sided

\((\overline{x}_1-\mu_2 ) - A_2 \leq \mu_1 - \mu_2 \leq (\overline{x}_1-\mu_2 ) + A_2\)

Type	Confidence Interval (Difference of Means)
Left-sided	\(-\infty \leq \mu_1 - \mu_2 \leq (\overline{x}_1-\mu_2) + A_1\)
Right-sided	\((\overline{x}_1-\mu_2 ) - A_1 \leq \mu_1 - \mu_2 \leq +\infty\)
Two-sided	\((\overline{x}_1-\mu_2 ) - A_2 \leq \mu_1 - \mu_2 \leq (\overline{x}_1-\mu_2 ) + A_2\)

An actual call to the function, requesting Student’s t-test with description, the critical value for a two-sided test, the p-value for \(H_{03}\) (in the case of textsf{TTest} this is \(\mu_1 \neq \mu_2\)), for 2 independent samples of size 10 and standard deviation 1 each, with means 2.3 and 4.5, and a type I error \(\alpha=0.05\) would be
>>> from mpfunlab import mpm
>>> xreal.StudentT1CI(mean:=5.24, mean0:=4.05, sd:=1.5, n:=22, alpha=0.05, resultstring)

Input
Variable Variable 1
Common N 22
Mean. Group 1 5.24
Mean. Group 2 4.05
StDev. Group 1 1.5
StDev. Group 2 3.5
Type 1 Error 0.05
Pearson's rho 0.7
Pearson's rho0. reference 0.2

Student's t-test for 2 corr. samples
Parameter Result
df 21
Difference of means 1.19
t-value (=delta) 2.087398086
t1 - alpha(1 - sided) 1.720742903
t1 - alpha(2 - sided) 2.079613845
µ1 - µ2. CI - Length (2 - sided) 2.371124599
µ1 - µ2. CI Upper Limit (2-sided) 2.3755623
µ1 - µ2. CI Lower Limit (2-sided) 0.0044377

Student t-test for 2 correlated samples: power#

ctx.studentt_2csamples_power(mean, sd, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns the results of Student’s t-test for 2 correlated samples: power and sample size.

Parameters:

Mean:: The mean of the sample.
Sd:: The standard deviation of the sample
N:: The sample size
Alpha:: The alpha-level used for confidence intervals

Let \(\sigma_1^2 = \sigma^2\) and \(\nu=N-1\). Define

\[\widetilde{\rho} = \frac{\mu_1-\mu_2}{\sigma} \text{ and } \delta = \sqrt{N} \widetilde{\rho}.\]

Let \(F_{t'}\left(\cdot, \nu, \delta \right)\) denote the CDF of the (singly) noncentral \(t\)-distribution with \(\nu\) degrees of freedom and noncentrality parameter \(\delta\) and let \(t_{\nu,\alpha}\) denote the \(\alpha\)-quantile of the central \(t\)-distribution with \(\nu\) degrees of freedom. Then the power for accepting \(H_A\) at the confidence level \(\alpha\) can be calculated as summarized below:

Test

Null Hypothesis

Alternative

Power

1 sided

\(H_{01}: \mu_1\leq \mu_2\)

\(H_{A1}: \mu_1> \mu_2\)

\(F_{t'}\left(-t_{\nu;1-\alpha}, \nu, \delta \right)\)

1 sided

\(H_{02}: \mu_1\geq \mu_2\)

\(H_{A2}: \mu_1< \mu_2\)

\(F_{t'}\left(t_{\nu;1-\alpha}, \nu, \delta \right)\)

2 sided

\(H_{03}: \mu_1= \mu_2\)

\(H_{A1}: \mu_1> \mu_2\)

\(F_{t'}\left(-t_{\nu;1-\alpha/2}, \nu, \delta \right)\)

2 sided

\(H_{03}: \mu_1= \mu_2\)

\(H_{A1}: \mu_1> \mu_2\)

\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)\)

2 sided

\(H_{03}: \mu_1= \mu_2\)

\(H_{A3}: \mu_1\neq \mu_2\)

\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)-F_t\left(-t_{\nu;1-\alpha/2}, \nu\, \delta \right)\)

Test	Null Hypothesis	Alternative	Power
1 sided	\(H_{01}: \mu_1\leq \mu_2\)	\(H_{A1}: \mu_1> \mu_2\)	\(F_{t'}\left(-t_{\nu;1-\alpha}, \nu, \delta \right)\)
1 sided	\(H_{02}: \mu_1\geq \mu_2\)	\(H_{A2}: \mu_1< \mu_2\)	\(F_{t'}\left(t_{\nu;1-\alpha}, \nu, \delta \right)\)
2 sided	\(H_{03}: \mu_1= \mu_2\)	\(H_{A1}: \mu_1> \mu_2\)	\(F_{t'}\left(-t_{\nu;1-\alpha/2}, \nu, \delta \right)\)
2 sided	\(H_{03}: \mu_1= \mu_2\)	\(H_{A1}: \mu_1> \mu_2\)	\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)\)
2 sided	\(H_{03}: \mu_1= \mu_2\)	\(H_{A3}: \mu_1\neq \mu_2\)	\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)-F_t\left(-t_{\nu;1-\alpha/2}, \nu\, \delta \right)\)

An actual call to the function, requesting Student’s t-test with description, the critical calue for a two-sided test, the power for \(H_{A3}\) (in the case of textsf{TTest} this is \(\mu_1 \neq \mu_2\)), for 2 independent samples of size 10 and standard deviation 1 each, with means 2.3 and 4.5, and a type I error \(\alpha=0.05\) would be

>>> from mpfunlab import mpm
>>> xreal.StudentT1CI(mu1:=5.24, mu0:=4.05, sd:=1.5, n:=22, alpha:=0.05)

Input
Variable Variable 1
Common N 22
Mean. Group 1 5.24
Mean. Group 2 4.05
StDev. Group 1 1.5
StDev. Group 2 3.5
Type 1 Error 0.05
Pearson's rho 0.7
Pearson's rho0. reference 0.2

Student's t-test for 2 corr. samples
Parameter Result
df 21
Difference of means 1.19
t-value (=delta) 2.087398086
t1 - alpha(1 - sided) 1.720742903
t1 - alpha(2 - sided) 2.079613845
1-sided test. power (HA1: µ1 < µ2) 0.000122287
1-sided test. power (HA2: µ1 > µ2) 0.646013238
2-sided test. power (HA1: µ1 < µ2) 3.78799E-05
2-sided test. power (HA2: µ1 > µ2) 0.512603358
2-sided test. power (HA3: µ1 <> µ2) 0.512641238
test. Pr[Mean 1 < Mean 2] 0.018426082
test. Pr[Mean 1 > Mean 2] 0.981573918

Student t-test for 2 correlated samples: sample size calculation#

ctx.studentt_2csamples_samplesize(mu, sd, alpha=0.05, beta=0.1)#

where ctx is dec, mpm, or gmp.

Returns the results of sample size calculations for Student’s t-test for 2 correlated samples

Parameters:

Mean:: The mean of the sample.
Sd:: The standard deviation of the sample
N:: The sample size
Alpha:: The alpha-level used for confidence intervals
Beta:: The beta-level used for power

Let \(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\) denote the sample size function of the (singly) noncentral \(t\)-distribution (see section ref{NoncentralTDistributionSampleSize} ) for a given confidence level \(\alpha\), power \(\beta\) and noncentrality parameter \(\widetilde{\rho}\) (as defined in equation ref{eq:TTestPower1}. The required total sample size \(N\) can be calculated as summarized below

Test

Null Hypothesis

Alternative

Minimal sample size

1 sided

\(H_{01}: \mu_1\leq \mu_2\)

\(H_{A1}: \mu_1> \mu_2\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

1 sided

\(H_{02}: \mu_1\geq \mu_2\)

\(H_{A2}: \mu_1< \mu_2\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

2 sided

\(H_{03}: \mu_1= \mu_2\)

\(H_{A1}: \mu_1> \mu_2\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

2 sided

\(H_{03}: \mu_1= \mu_2\)

\(H_{A1}: \mu_1> \mu_2\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

2 sided

\(H_{03}: \mu_1= \mu_2\)

\(H_{A3}: \mu_1\neq \mu_2\)

\(N2_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

Test	Null Hypothesis	Alternative	Minimal sample size
1 sided	\(H_{01}: \mu_1\leq \mu_2\)	\(H_{A1}: \mu_1> \mu_2\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
1 sided	\(H_{02}: \mu_1\geq \mu_2\)	\(H_{A2}: \mu_1< \mu_2\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
2 sided	\(H_{03}: \mu_1= \mu_2\)	\(H_{A1}: \mu_1> \mu_2\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
2 sided	\(H_{03}: \mu_1= \mu_2\)	\(H_{A1}: \mu_1> \mu_2\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
2 sided	\(H_{03}: \mu_1= \mu_2\)	\(H_{A3}: \mu_1\neq \mu_2\)	\(N2_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

Note that the returned value of \(N\) will in general not be an integer, and rounding up may be required.

An actual call to the function, requesting an upper sample size estimate (and actual power) for \(\alpha = 0.95\), \(\beta=0.1\) , and standard deviations \(\sigma_1=\sigma_2=1\) , means \(\mu_1=2.3\) and \(\mu_2=4.5\), would be

>>> from mpfunlab import mpm
>>> xreal.StudentT1CI(mu1:=5.24, mu0:=4.05, sd:=1.5, alpha:=0.05, beta:=0.1)
                                         df: 21
                        difference of means: 1.19
                           t-value (=delta): 3.721063
                        t(1-alpha, 1-sided): 1.720743
                        t(1-alpha, 2-sided): 2.079614
  1-sided test, required N (HA1: mu1 < mu2): 18
1-sided test, actual power (HA1: mu1 < mu2): 0.974564
  1-sided test, required N (HA2: mu1 > mu2): 148
1-sided test, actual power (HA2: mu1 > mu2): 0.964564
  2-sided test, required N (HA1: mu1 < mu2): 22
2-sided test, actual power (HA1: mu1 < mu2): 0.954564
  2-sided test, required N (HA2: mu1 > mu2): 212
2-sided test, actual power (HA2: mu1 > mu2): 0.977456
  2-sided test, required N (HA2: mu1 <>mu2): 24
2-sided test, actual power (HA2: mu1 <>mu2): 0.955544

Morgan-Pitman test for the variances of 2 correlated samples: tests (p-values)#

ctx.fratio_variance_2csamples_test(s2, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns results for the Morgan-Pitman test for the variances of 2 correlated samples.

Parameters:

Sd:: The standard deviation of the sample
N:: The sample size
Alpha:: The alpha-level used for confidence intervals

Let \((X_1, X_2, \ldots, X_N)\) denote a random sample of size \(N\) from a normal distribution with mean \(\sigma\) and variance \(\sigma^2\), and let

\[\overline{x}_1 = \frac{1}{N} \sum_{i=1}^N X_i \quad \text{and } s^2 = \frac{1}{N-1} \sum_{i=1}^N (X_i - \overline{x}_1)\]

be the usual sample estimates of the unkown population mean \(\sigma\) and unkown population variance \(\sigma^2\). Then Student’s t-test can be used to test hypotheses concerning \(\sigma\) with regard to a reference value \(\sigma_2\).

\[t= \frac{\overline{x}_1-\sigma_2}{s}, \quad s=\sqrt{s_1^2 /N}, \quad \nu=N-1.\]

Then \(p\)-values and rejection criteria for \(H_0\) can be calculated as summarized below

Test problem

\(p\)-value

Reject \(H_0\)

\(H_{01}: \sigma_1\leq \sigma_2\) vs \(H_{A1}: \sigma_1> \sigma_2\)

\(F_t\left(-t, \nu\right)\)

\(t > t_{\nu;1-\alpha}\)

\(H_{02}: \sigma_1\geq \sigma_2\) vs \(H_{A2}: \sigma_1< \sigma_2\)

\(F_t\left(t, \nu\right)\)

\(t > t_{\nu;\alpha}\)

\(H_{03}: \sigma_1= \sigma_2\) vs \(H_{A3}: \sigma_1\neq \sigma_2\)

\(F_t\left(t, \nu\right)-F_t\left(-t, \nu\right)\)

\(t > t_{\nu;1-\alpha/2}\) or \(t > t_{\nu;\alpha/2}\)

Test problem	\(p\)-value	Reject \(H_0\)
\(H_{01}: \sigma_1\leq \sigma_2\) vs \(H_{A1}: \sigma_1> \sigma_2\)	\(F_t\left(-t, \nu\right)\)	\(t > t_{\nu;1-\alpha}\)
\(H_{02}: \sigma_1\geq \sigma_2\) vs \(H_{A2}: \sigma_1< \sigma_2\)	\(F_t\left(t, \nu\right)\)	\(t > t_{\nu;\alpha}\)
\(H_{03}: \sigma_1= \sigma_2\) vs \(H_{A3}: \sigma_1\neq \sigma_2\)	\(F_t\left(t, \nu\right)-F_t\left(-t, \nu\right)\)	\(t > t_{\nu;1-\alpha/2}\) or \(t > t_{\nu;\alpha/2}\)

The test can also be expressed in terms of a correlation coefficient \(r\) between the combined \(X\) and an indicator variable, where \(t\) and \(r\) are related by

\[r=\frac{t}{\sqrt{t^2+\nu}}, \quad t= \nu \frac{r}{1-r^2}.\]

An actual call to the function, requesting Student’s t-test with description, the critical value for a two-sided test, the p-value for \(H_{03}\) (in the case of \(\textsf{TTest}\) this is \(\sigma_1 \neq \sigma_2\)), for 2 independent samples of size 10 and standard deviation 1 each, with means 2.3 and 4.5, and a type I error \(\alpha=0.05\) would be
>>> from mpfunlab import mpm
>>> xreal.StudentT1Test(means:=[5.24, 4.05], sd:=1.5, n:=22)

Input
Variable Variable 1
Common N 22
Mean. Group 1 5.24
Mean. Group 2 4.05
StDev. Group 1 1.5
StDev. Group 2 3.5
Pearson's rho 0.7
Pearson's rho0. reference 0.2

Pitman-Morgan-test for 2 corr. variances
Parameter Result
df1 21
df2 21
Variance-Ratio 0.183673469
F. 1 - alpha(1 - sided) 2.084188623
F . 1 - alpha(2 - sided) 2.408589482
Pitman-Morgan F-test. p-value (H01: s1 >= s2) 0.999864065
Pitman-Morgan F-test. p-value (H02: s1 <= s2) 0.000135935
Pitman-Morgan F-test. p-value (H03: s1 = s2) 0.999728131

Morgan-Pitman test for the variances of 2 correlated samples: confidence intervals#

ctx.fratio_variance_2csamples_ci(s2, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns results of the confidence intervals for the Morgan-Pitman test for the variances of 2 correlated samples.

Parameters:

Mean:: The mean of the sample.
Sd:: The standard deviation of the sample
N:: The sample size
Alpha:: The alpha-level used for confidence intervals

Type

Confidence Interval (Difference of Means)

Left-sided

\(-\infty \leq \sigma_1 - \sigma_2 \leq (\overline{x}_1-\sigma_2) + A_1\)

Right-sided

\((\overline{x}_1-\sigma_2 ) - A_1 \leq \sigma_1 - \sigma_2 \leq +\infty\)

Two-sided

\((\overline{x}_1-\sigma_2 ) - A_2 \leq \sigma_1 - \sigma_2 \leq (\overline{x}_1-\sigma_2 ) + A_2\)

Type	Confidence Interval (Difference of Means)
Left-sided	\(-\infty \leq \sigma_1 - \sigma_2 \leq (\overline{x}_1-\sigma_2) + A_1\)
Right-sided	\((\overline{x}_1-\sigma_2 ) - A_1 \leq \sigma_1 - \sigma_2 \leq +\infty\)
Two-sided	\((\overline{x}_1-\sigma_2 ) - A_2 \leq \sigma_1 - \sigma_2 \leq (\overline{x}_1-\sigma_2 ) + A_2\)

An actual call to the function, requesting Student’s t-test with description, the critical value for a two-sided test, the p-value for \(H_{03}\) (in the case of textsf{TTest} this is \(\sigma_1 \neq \sigma_2\)), for 2 independent samples of size 10 and standard deviation 1 each, with means 2.3 and 4.5, and a type I error \(\alpha=0.05\) would be
>>> from mpfunlab import mpm
>>> xreal.StudentT1CI(mean:=5.24, mean0:=4.05, sd:=1.5, n:=22, alpha=0.05, resultstring)

Input
Variable Variable 1
Common N 22
Mean. Group 1 5.24
Mean. Group 2 4.05
StDev. Group 1 1.5
StDev. Group 2 3.5
Type 1 Error 0.05
Pearson's rho 0.7
Pearson's rho0. reference 0.2

Pitman-Morgan-test for 2 corr. variances
Parameter Result
df1 21
df2 21
Variance-Ratio 0.183673469
F. 1 - alpha(1 - sided) 2.084188623
F . 1 - alpha(2 - sided) 2.408589482
s1/s2. CI - Length(2 - sided) 0.366136297
s1/s2. CI Upper Limit (2-sided) 0.442393986
s1/s2. CI Lower Limit (2-sided) 0.07625769

Morgan-Pitman test for the variances of 2 correlated samples: power#

ctx.fratio_variance_2csamples_power(s2, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns results of the power calculation for the Morgan-Pitman test for the variances of 2 correlated samples.

Parameters:

Sd:: The standard deviation of the sample
N:: The sample size
Alpha:: The alpha-level used for confidence intervals

Let \(\sigma_1^2 = \sigma^2\) and \(\nu=N-1\). Define

\[\widetilde{\rho} = \frac{\sigma_1-\sigma_2}{\sigma} \text{ and } \delta = \sqrt{N} \widetilde{\rho}.\]

Test

Null Hypothesis

Alternative

Power

1 sided

\(H_{01}: \sigma_1\leq \sigma_2\)

\(H_{A1}: \sigma_1> \sigma_2\)

\(F_{t'}\left(-t_{\nu;1-\alpha}, \nu, \delta \right)\)

1 sided

\(H_{02}: \sigma_1\geq \sigma_2\)

\(H_{A2}: \sigma_1< \sigma_2\)

\(F_{t'}\left(t_{\nu;1-\alpha}, \nu, \delta \right)\)

2 sided

\(H_{03}: \sigma_1= \sigma_2\)

\(H_{A1}: \sigma_1> \sigma_2\)

\(F_{t'}\left(-t_{\nu;1-\alpha/2}, \nu, \delta \right)\)

2 sided

\(H_{03}: \sigma_1= \sigma_2\)

\(H_{A1}: \sigma_1> \sigma_2\)

\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)\)

2 sided

\(H_{03}: \sigma_1= \sigma_2\)

\(H_{A3}: \sigma_1\neq \sigma_2\)

\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)-F_t\left(-t_{\nu;1-\alpha/2}, \nu\, \delta \right)\)

Test	Null Hypothesis	Alternative	Power
1 sided	\(H_{01}: \sigma_1\leq \sigma_2\)	\(H_{A1}: \sigma_1> \sigma_2\)	\(F_{t'}\left(-t_{\nu;1-\alpha}, \nu, \delta \right)\)
1 sided	\(H_{02}: \sigma_1\geq \sigma_2\)	\(H_{A2}: \sigma_1< \sigma_2\)	\(F_{t'}\left(t_{\nu;1-\alpha}, \nu, \delta \right)\)
2 sided	\(H_{03}: \sigma_1= \sigma_2\)	\(H_{A1}: \sigma_1> \sigma_2\)	\(F_{t'}\left(-t_{\nu;1-\alpha/2}, \nu, \delta \right)\)
2 sided	\(H_{03}: \sigma_1= \sigma_2\)	\(H_{A1}: \sigma_1> \sigma_2\)	\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)\)
2 sided	\(H_{03}: \sigma_1= \sigma_2\)	\(H_{A3}: \sigma_1\neq \sigma_2\)	\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)-F_t\left(-t_{\nu;1-\alpha/2}, \nu\, \delta \right)\)

An actual call to the function, requesting Student’s t-test with description, the critical calue for a two-sided test, the power for \(H_{A3}\) (in the case of textsf{TTest} this is \(\sigma_1 \neq \sigma_2\)), for 2 independent samples of size 10 and standard deviation 1 each, with means 2.3 and 4.5, and a type I error \(\alpha=0.05\) would be

>>> from mpfunlab import mpm
>>> xreal.StudentT1CI(mu1:=5.24, mu0:=4.05, sd:=1.5, n:=22, alpha:=0.05)

Input
Variable Variable 1
Common N 22
Mean. Group 1 5.24
Mean. Group 2 4.05
StDev. Group 1 1.5
StDev. Group 2 3.5
Type 1 Error 0.05
Pearson's rho 0.7
Pearson's rho0. reference 0.2

Pitman-Morgan-test for 2 corr. variances
Parameter Result
df1 21
df2 21
Variance-Ratio 0.183673469
F. 1 - alpha(1 - sided) 2.084188623
F . 1 - alpha(2 - sided) 2.408589482
Pitman-Morgan 1-sided test. power (HA1: s1 < s2) 0.999864065
Pitman-Morgan 1-sided test. power (HA2: s1 > s2) 0.999864065
Pitman-Morgan 2-sided test. power (HA1: s1 < s2) 0.999864065
Pitman-Morgan 2-sided test. power (HA2: s1 > s2) 0.999864065
Pitman-Morgan 2-sided test. power (HA3: s1 <> s2) 0.999864065

Morgan-Pitman test for the variances of 2 correlated samples: sample size#

ctx.fratio_variance_2csamples_samplesize(s2, alpha=0.05, beta=0.1)#

where ctx is dec, mpm, or gmp.

Returns results of sample size calculations for the Morgan-Pitman test for the variances of 2 correlated samples.

Parameters:

Sd:: The standard deviation of the sample
Alpha:: The alpha-level used for confidence intervals
Beta:: The beta-level used for power

Test

Null Hypothesis

Alternative

Minimal sample size

1 sided

\(H_{01}: \sigma_1\leq \sigma_2\)

\(H_{A1}: \sigma_1> \sigma_2\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

1 sided

\(H_{02}: \sigma_1\geq \sigma_2\)

\(H_{A2}: \sigma_1< \sigma_2\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

2 sided

\(H_{03}: \sigma_1= \sigma_2\)

\(H_{A1}: \sigma_1> \sigma_2\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

2 sided

\(H_{03}: \sigma_1= \sigma_2\)

\(H_{A1}: \sigma_1> \sigma_2\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

2 sided

\(H_{03}: \sigma_1= \sigma_2\)

\(H_{A3}: \sigma_1\neq \sigma_2\)

\(N2_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

Test	Null Hypothesis	Alternative	Minimal sample size
1 sided	\(H_{01}: \sigma_1\leq \sigma_2\)	\(H_{A1}: \sigma_1> \sigma_2\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
1 sided	\(H_{02}: \sigma_1\geq \sigma_2\)	\(H_{A2}: \sigma_1< \sigma_2\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
2 sided	\(H_{03}: \sigma_1= \sigma_2\)	\(H_{A1}: \sigma_1> \sigma_2\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
2 sided	\(H_{03}: \sigma_1= \sigma_2\)	\(H_{A1}: \sigma_1> \sigma_2\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
2 sided	\(H_{03}: \sigma_1= \sigma_2\)	\(H_{A3}: \sigma_1\neq \sigma_2\)	\(N2_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

Note that the returned value of \(N\) will in general not be an integer, and rounding up may be required.

>>> from mpfunlab import mpm
>>> xreal.StudentT1CI(mu1:=5.24, mu0:=4.05, sd:=1.5, alpha:=0.05, beta:=0.1)
                                         df: 21
                        difference of means: 1.19
                           t-value (=delta): 3.721063
                        t(1-alpha, 1-sided): 1.720743
                        t(1-alpha, 2-sided): 2.079614
  1-sided test, required N (HA1: mu1 < mu2): 18
1-sided test, actual power (HA1: mu1 < mu2): 0.974564
  1-sided test, required N (HA2: mu1 > mu2): 148
1-sided test, actual power (HA2: mu1 > mu2): 0.964564
  2-sided test, required N (HA1: mu1 < mu2): 22
2-sided test, actual power (HA1: mu1 < mu2): 0.954564
  2-sided test, required N (HA2: mu1 > mu2): 212
2-sided test, actual power (HA2: mu1 > mu2): 0.977456
  2-sided test, required N (HA2: mu1 <>mu2): 24
2-sided test, actual power (HA2: mu1 <>mu2): 0.955544

Pearson’s rho, 2 correlated samples: tests (p-values)#

ctx.pearson_rho_test(rho, rh0, n, alpha=0.05, rtype=1)#

where ctx is dec, mpm, or gmp.

Returns results of tests for Pearson’s rho, 2 correlated samples.

Parameters:

Mean:: The mean of the sample.
Mean0:: The reference mean.
Sd:: The standard deviation of the sample
N:: The sample size
Alpha:: The alpha-level used for confidence intervals

Let \((X_1, X_2, \ldots, X_N)\) denote a random sample of size \(N\) from a normal distribution with mean \(\rho\) and variance \(\sigma^2\), and let

\[\overline{x}_1 = \frac{1}{N} \sum_{i=1}^N X_i \quad \text{and } s^2 = \frac{1}{N-1} \sum_{i=1}^N (X_i - \overline{x}_1)\]

be the usual sample estimates of the unkown population mean \(\rho\) and unkown population variance \(\sigma^2\). Then Student’s t-test can be used to test hypotheses concerning \(\rho\) with regard to a reference value \(\rho_0\).

\[t= \frac{\overline{x}_1-\rho_0}{s}, \quad s=\sqrt{s_1^2 /N}, \quad \nu=N-1.\]

Then \(p\)-values and rejection criteria for \(H_0\) can be calculated as summarized below

Test problem

\(p\)-value

Reject \(H_0\)

\(H_{01}: \rho \leq \rho_0\) vs \(H_{A1}: \rho > \rho_0\)

\(F_t\left(-t, \nu\right)\)

\(t > t_{\nu;1-\alpha}\)

\(H_{02}: \rho \geq \rho_0\) vs \(H_{A2}: \rho < \rho_0\)

\(F_t\left(t, \nu\right)\)

\(t > t_{\nu;\alpha}\)

\(H_{03}: \rho = \rho_0\) vs \(H_{A3}: \rho \neq \rho_0\)

\(F_t\left(t, \nu\right)-F_t\left(-t, \nu\right)\)

\(t > t_{\nu;1-\alpha/2}\) or \(t > t_{\nu;\alpha/2}\)

Test problem	\(p\)-value	Reject \(H_0\)
\(H_{01}: \rho \leq \rho_0\) vs \(H_{A1}: \rho > \rho_0\)	\(F_t\left(-t, \nu\right)\)	\(t > t_{\nu;1-\alpha}\)
\(H_{02}: \rho \geq \rho_0\) vs \(H_{A2}: \rho < \rho_0\)	\(F_t\left(t, \nu\right)\)	\(t > t_{\nu;\alpha}\)
\(H_{03}: \rho = \rho_0\) vs \(H_{A3}: \rho \neq \rho_0\)	\(F_t\left(t, \nu\right)-F_t\left(-t, \nu\right)\)	\(t > t_{\nu;1-\alpha/2}\) or \(t > t_{\nu;\alpha/2}\)

The test can also be expressed in terms of a correlation coefficient \(r\) between the combined \(X\) and an indicator variable, where \(t\) and \(r\) are related by

\[r=\frac{t}{\sqrt{t^2+\nu}}, \quad t= \nu \frac{r}{1-r^2}.\]

An actual call to the function, requesting Student’s t-test with description, the critical value for a two-sided test, the p-value for \(H_{03}\) (in the case of \(\textsf{TTest}\) this is \(\rho_1 \neq \rho_2\)), for 2 independent samples of size 10 and standard deviation 1 each, with means 2.3 and 4.5, and a type I error \(\alpha=0.05\) would be
>>> from mpfunlab import mpm
>>> xreal.StudentT1Test(mean:=5.24, mean0:=4.05, sd:=1.5, n:=22, resultstring:='All')

Input
VariableVariable1
CommonN22
Mean.Group15.24
Mean.Group24.05
StDev.Group11.5
StDev.Group23.5
Pearson'srho0.7
Pearson'srho0.reference0.2

Pearson'srho
ParameterResult
df
rho0.1-alpha(1-sided)
rho0.1-alpha(2-sided)
p-value(H01:rho>=0)
p-value(H02:rho<=0)
p-value(H03:rho=0)
p-value(H01:rho>=rho0)
p-value(H02:rho<=rho0)
p-value(H03:rho=rho0)
p-value(H04:rho^2=rho0^2)

Pearson’s rho, 2 correlated samples: confidence intervals#

ctx.pearson_rho_ci(rho, rh0, n, alpha=0.05, rtype=1)#

where ctx is dec, mpm, or gmp.

Returns results of confidence intervals for Pearson’s rho, 2 correlated samples.

Parameters:

Mean:: The mean of the sample.
Mean0:: The reference mean.
Sd:: The standard deviation of the sample
N:: The sample size
Alpha:: The alpha-level used for confidence intervals

Type

Confidence Interval (Difference of Means)

Left-sided

\(-\infty \leq \rho_1 - \rho_0 \leq (\overline{x}_1-\rho_0) + A_1\)

Right-sided

\((\overline{x}_1-\rho_0 ) - A_1 \leq \rho_1 - \rho_0 \leq +\infty\)

Two-sided

\((\overline{x}_1-\rho_0 ) - A_2 \leq \rho_1 - \rho_0 \leq (\overline{x}_1-\rho_0 ) + A_2\)

Type	Confidence Interval (Difference of Means)
Left-sided	\(-\infty \leq \rho_1 - \rho_0 \leq (\overline{x}_1-\rho_0) + A_1\)
Right-sided	\((\overline{x}_1-\rho_0 ) - A_1 \leq \rho_1 - \rho_0 \leq +\infty\)
Two-sided	\((\overline{x}_1-\rho_0 ) - A_2 \leq \rho_1 - \rho_0 \leq (\overline{x}_1-\rho_0 ) + A_2\)

An actual call to the function, requesting Student’s t-test with description, the critical value for a two-sided test, the p-value for \(H_{03}\) (in the case of textsf{TTest} this is \(\rho_1 \neq \rho_2\)), for 2 independent samples of size 10 and standard deviation 1 each, with means 2.3 and 4.5, and a type I error \(\alpha=0.05\) would be
>>> from mpfunlab import mpm
>>> xreal.StudentT1CI(mean:=5.24, mean0:=4.05, sd:=1.5, n:=22, alpha=0.05, resultstring)

Input
Variable Variable 1
Common N 22
Mean. Group 1 5.24
Mean. Group 2 4.05
StDev. Group 1 1.5
StDev. Group 2 3.5
Type 1 Error 0.05
Pearson's rho 0.7
Pearson's rho0. reference 0.2

Pearson's rho
Parameter Result
df
rho0. 1 - alpha(1 - sided)
rho0 . 1 - alpha(2 - sided)
rho. CI Upper Limit (2-sided)
rho. estimate
rho. CI Lower Limit (2-sided)
rho. CI - Length(2 - sided)

Pearson’s rho, 2 correlated samples:: power#

ctx.pearson_rho_power(rho, rho0, n, alpha=0.05, rtype=1)#

where ctx is dec, mpm, or gmp.

Returns the results of power calculations for Pearson’s rho, 2 correlated samples.

Parameters:

Rho:: The mean of the sample.
Rho0:: The reference mean.
N:: The sample size
Alpha:: The alpha-level used for confidence intervals

Let \(\sigma_1^2 = \sigma^2\) and \(\nu=N-1\). Define

\[\widetilde{\rho} = \frac{\rho_1-\rho_0}{\sigma} \text{ and } \delta = \sqrt{N} \widetilde{\rho}.\]

Test

Null Hypothesis

Alternative

Power

1 sided

\(H_{01}: \rho \leq \rho_0\)

\(H_{A1}: \rho > \rho_0\)

\(F_{t'}\left(-t_{\nu;1-\alpha}, \nu, \delta \right)\)

1 sided

\(H_{02}: \rho \geq \rho_0\)

\(H_{A2}: \rho < \rho_0\)

\(F_{t'}\left(t_{\nu;1-\alpha}, \nu, \delta \right)\)

2 sided

\(H_{03}: \rho = \rho_0\)

\(H_{A1}: \rho > \rho_0\)

\(F_{t'}\left(-t_{\nu;1-\alpha/2}, \nu, \delta \right)\)

2 sided

\(H_{03}: \rho = \rho_0\)

\(H_{A1}: \rho > \rho_0\)

\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)\)

2 sided

\(H_{03}: \rho = \rho_0\)

\(H_{A3}: \rho \neq \rho_0\)

\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)-F_t\left(-t_{\nu;1-\alpha/2}, \nu\, \delta \right)\)

Test	Null Hypothesis	Alternative	Power
1 sided	\(H_{01}: \rho \leq \rho_0\)	\(H_{A1}: \rho > \rho_0\)	\(F_{t'}\left(-t_{\nu;1-\alpha}, \nu, \delta \right)\)
1 sided	\(H_{02}: \rho \geq \rho_0\)	\(H_{A2}: \rho < \rho_0\)	\(F_{t'}\left(t_{\nu;1-\alpha}, \nu, \delta \right)\)
2 sided	\(H_{03}: \rho = \rho_0\)	\(H_{A1}: \rho > \rho_0\)	\(F_{t'}\left(-t_{\nu;1-\alpha/2}, \nu, \delta \right)\)
2 sided	\(H_{03}: \rho = \rho_0\)	\(H_{A1}: \rho > \rho_0\)	\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)\)
2 sided	\(H_{03}: \rho = \rho_0\)	\(H_{A3}: \rho \neq \rho_0\)	\(F_{t'}\left(t_{\nu;1-\alpha/2}, \nu, \delta \right)-F_t\left(-t_{\nu;1-\alpha/2}, \nu\, \delta \right)\)

An actual call to the function, requesting Student’s t-test with description, the critical calue for a two-sided test, the power for \(H_{A3}\) (in the case of textsf{TTest} this is \(\rho_1 \neq \rho_2\)), for 2 independent samples of size 10 and standard deviation 1 each, with means 2.3 and 4.5, and a type I error \(\alpha=0.05\) would be
>>> from mpfunlab import mpm
>>> xreal.StudentT1CI(mu1:=5.24, mu0:=4.05, sd:=1.5, n:=22, alpha:=0.05)


Input
Variable Variable 1
Common N 22
Mean. Group 1 5.24
Mean. Group 2 4.05
StDev. Group 1 1.5
StDev. Group 2 3.5
Type 1 Error 0.05
Pearson's rho 0.7
Pearson's rho0. reference 0.2

Pearson's rho
Parameter Result
df
rho0. 1 - alpha(1 - sided)
rho0 . 1 - alpha(2 - sided)
Power (HA1: rho > rho0)
Power (HA2: rho < rho0)
Power (HA3: rho <> rho0)
Power (HA4: rho^2 <> rho0^2)

Pearson’s rho, 2 correlated samples:: sample size calculation#

ctx.pearson_rho_samplesize(rho, rho0, n, alpha=0.05, beta=0.1, rtype=1)#

where ctx is dec, mpm, or gmp.

Returns the results of power calculations for Pearson’s rho, 2 correlated samples.

Parameters:

Rho:: The mean of the sample.
Rho0:: The reference mean.
N:: The sample size
Alpha:: The alpha-level used for confidence intervals
Beta:: The beta-level used for power

Test

Null Hypothesis

Alternative

Minimal sample size

1 sided

\(H_{01}: \rho \leq \rho_0\)

\(H_{A1}: \rho > \rho_0\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

1 sided

\(H_{02}: \rho \geq \rho_0\)

\(H_{A2}: \rho < \rho_0\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

2 sided

\(H_{03}: \rho = \rho_0\)

\(H_{A1}: \rho > \rho_0\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

2 sided

\(H_{03}: \rho = \rho_0\)

\(H_{A1}: \rho > \rho_0\)

\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

2 sided

\(H_{03}: \rho = \rho_0\)

\(H_{A3}: \rho \neq \rho_0\)

\(N2_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

Test	Null Hypothesis	Alternative	Minimal sample size
1 sided	\(H_{01}: \rho \leq \rho_0\)	\(H_{A1}: \rho > \rho_0\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
1 sided	\(H_{02}: \rho \geq \rho_0\)	\(H_{A2}: \rho < \rho_0\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
2 sided	\(H_{03}: \rho = \rho_0\)	\(H_{A1}: \rho > \rho_0\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
2 sided	\(H_{03}: \rho = \rho_0\)	\(H_{A1}: \rho > \rho_0\)	\(N_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)
2 sided	\(H_{03}: \rho = \rho_0\)	\(H_{A3}: \rho \neq \rho_0\)	\(N2_{t'}\left(\alpha, \beta, \widetilde{\rho} \right)\)

Note that the returned value of \(N\) will in general not be an integer, and rounding up may be required.

>>> from mpfunlab import mpm
>>> xreal.StudentT1CI(mu1:=5.24, mu0:=4.05, sd:=1.5, alpha:=0.05, beta:=0.1)
                                         df: 21
                        difference of means: 1.19
                           t-value (=delta): 3.721063
                        t(1-alpha, 1-sided): 1.720743
                        t(1-alpha, 2-sided): 2.079614
  1-sided test, required N (HA1: mu1 < mu2): 18
1-sided test, actual power (HA1: mu1 < mu2): 0.974564
  1-sided test, required N (HA2: mu1 > mu2): 148
1-sided test, actual power (HA2: mu1 > mu2): 0.964564
  2-sided test, required N (HA1: mu1 < mu2): 22
2-sided test, actual power (HA1: mu1 < mu2): 0.954564
  2-sided test, required N (HA2: mu1 > mu2): 212
2-sided test, actual power (HA2: mu1 > mu2): 0.977456
  2-sided test, required N (HA2: mu1 <>mu2): 24
2-sided test, actual power (HA2: mu1 <>mu2): 0.955544

Basic classical statistical tests for 2 correlated samples (stratified)

Contents

Basic classical statistical tests for 2 correlated samples (stratified)#

Student t-test for 2 correlated samples: tests (p-values)#

Student t-test for 2 correlated samples: confidence intervals#

Student t-test for 2 correlated samples: power#

Student t-test for 2 correlated samples: sample size calculation#

Morgan-Pitman test for the variances of 2 correlated samples: tests (p-values)#

Morgan-Pitman test for the variances of 2 correlated samples: confidence intervals#

Morgan-Pitman test for the variances of 2 correlated samples: power#

Morgan-Pitman test for the variances of 2 correlated samples: sample size#

Pearson’s rho, 2 correlated samples: tests (p-values)#

Pearson’s rho, 2 correlated samples: confidence intervals#

Pearson’s rho, 2 correlated samples:: power#

Pearson’s rho, 2 correlated samples:: sample size calculation#