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

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

ctx.studentt_2isamples_test(mean, sd, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns results for Student’s t-test for for 2 independent 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

Student's t-test for 2 samples
Parameter Result
df 42
Difference of means 1.19
t-value (=delta) 1.465798594
t1 - alpha(1 - sided) 1.681952357
t1 - alpha(2 - sided) 2.018081703
test. p-value (H01: µ1 >= µ2) 0.924924828
test. p-value (H02: µ1 <= µ2) 0.075075172
test. p-value (H03: µ1 = µ2) 0.150150344

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

ctx.studentt_2isamples_ci(mean, sd, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns results for Student’s t-test for for 2 independent 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

Student's t-test for 2 samples
Parameter Result
df 42
Difference of means 1.19
t-value (=delta) 1.465798594
t1 - alpha(1 - sided) 1.681952357
t1 - alpha(2 - sided) 2.018081703
µ1 - µ2. CI - Length (2 - sided) 3.276735613
µ1 - µ2. CI Upper Limit (2-sided) 2.828367806
µ1 - µ2. CI Lower Limit (2-sided) -0.448367806

Student t-test for 2 independent samples: power#

ctx.studentt_2isamples_power(mean, sd, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns the results of Student’s t-test for 2 independent 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

Student's t-test for 2 samples
Parameter Result
df 42
Difference of means 1.19
t-value (=delta) 1.465798594
t1 - alpha(1 - sided) 1.681952357
t1 - alpha(2 - sided) 2.018081703
1-sided test. power (HA1: µ1 < µ2) 0.001009403
1-sided test. power (HA2: µ1 > µ2) 0.419683559
2-sided test. power (HA1: µ1 < µ2) 0.000345576
2-sided test. power (HA2: µ1 > µ2) 0.298885242
2-sided test. power (HA3: µ1 <> µ2) 0.299230817
test. Pr[Mean 1 < Mean 2] 0.071351582
test. Pr[Mean 1 > Mean 2] 0.928648418

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

ctx.studentt_2isamples_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 independent 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

Student t-test for 2 independent samples: power and sample size: failure to stratify#

ctx.studentt_2isamples_power2(mean, sd, n, alpha=0.05)#

Returns the results of the F-Test with \(\delta>0\) and \(\eta>0\)

See “Butler Paolella 1999 Doubly noncentral F Draft.pdf” in “References B” for an example.

Student t-test for 2 independent samples: equivalence and non-inferiority#

ctx.studentt_2isamples_equivalence(x, k, n, method='default')#

where ctx is dec, mpm, or gmp.

Returns the results of the Sign test, under \(H_0\).

Refer to Schuirman procedure.

See Shieh 2019

F-test for the variances of 2 independent samples: tests (p-values)#

ctx.fratio_variance_2isamples_test(s2, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns results for the F-test for the variances of 2 independent 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

F-test for 2 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
F-test. p-value (H01: s1 >= s2) 0.999864065
F-test. p-value (H02: s1 <= s2) 0.000135935
F-test. p-value (H03: s1 = s2) 1.999728131

F-test for the variances of 2 independent samples: confidence intervals#

ctx.fratio_variance_2isamples_ci(s2, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns results of the confidence intervals for the F-test for the variances of 2 independent 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

F-test for 2 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

F-test for the variances of 2 independent samples: power#

ctx.fratio_variance_2isamples_power(s2, n, alpha=0.05)#

where ctx is dec, mpm, or gmp.

Returns results of the power calculation for the F-test for the variances of 2 independent 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

F-test for 2 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
1-sided test. power (HA1: s1 < s2) 0.999864065
1-sided test. power (HA2: s1 > s2) 0.999864065
2-sided test. power (HA1: s1 < s2) 0.999864065
2-sided test. power (HA2: s1 > s2) 0.999864065
2-sided test. power (HA3: s1 <> s2) 0.999864065

F-test for the variances of 2 independent samples: sample size#

ctx.fratio_variance_2isamples_samplesize(s2, alpha=0.05, beta=0.1)#

where ctx is dec, mpm, or gmp.

Returns results of sample size calculations for the variances of 2 independent 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

Basic classical statistical tests for 2 independent samples (stratified)

Contents

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

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

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

Student t-test for 2 independent samples: power#

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

Student t-test for 2 independent samples: power and sample size: failure to stratify#

Student t-test for 2 independent samples: equivalence and non-inferiority#

F-test for the variances of 2 independent samples: tests (p-values)#

F-test for the variances of 2 independent samples: confidence intervals#

F-test for the variances of 2 independent samples: power#

F-test for the variances of 2 independent samples: sample size#