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|newpage| Multivariate statistical tests =============================================================== Canonical correlation: Canonical correlation: See also: Wikipedia :cite:p:`WikipediaStat300`. Linear_discriminant_analysis: See also: Wikipedia :cite:p:`WikipediaStat310`. Principal_component_analysis: See also: Wikipedia :cite:p:`WikipediaStat320`. Regression_analysis: See also: Wikipedia :cite:p:`WikipediaStat330`. Linear_regression: See also: Wikipedia :cite:p:`WikipediaStat340`. Analysis_of_variance: See also: Wikipedia :cite:p:`WikipediaStat350`. One-way_analysis_of_variance: See also: Wikipedia :cite:p:`WikipediaStat360`. Design_matrix: See also: Wikipedia :cite:p:`WikipediaStat370`. Analysis_of_covariance: See also: Wikipedia :cite:p:`WikipediaStat380`. General_linear_model: See also: Wikipedia :cite:p:`WikipediaStat390`. Multivariate_analysis_of_variance: See also: Wikipedia :cite:p:`WikipediaStat400`. Multivariate_analysis_of_covariance: See also: Wikipedia :cite:p:`WikipediaStat410`. Correlation_and_dependence: See also: Wikipedia :cite:p:`WikipediaStat420`. Standardization: See also: Wikipedia :cite:p:`WikipediaStat430`. Multiple linear regression: p-value and confidence interval ------------------------------------------------------------------------------------ .. method:: ctx.multlinreg_test(self, ctx, mean, mean0, sd, n, alpha=0.05) where ``ctx`` is ``dec``, ``mpm``, or ``gmp``. Returns the results of tests for a CR or RB Anova. Multiple linear regression Type I: power and sample size --------------------------------------------------------------------------------- .. method:: ctx.multlinreg_type1_power(self, ctx, mean, mean0, sd, n, alpha=0.05) where ``ctx`` is ``dec``, ``mpm``, or ``gmp``. Returns the results of tests for a CR or RB Anova. Multiple linear regression Type II: power and sample size ------------------------------------------------------------------------------------ .. method:: ctx.multlinreg_type2_power(self, ctx, mean, mean0, sd, n, alpha=0.05) where ``ctx`` is ``dec``, ``mpm``, or ``gmp``. Returns the results of tests for a CR or RB Anova. Ref: Kelley (2008): Sample size, confidence interval |newpage| Hotelling's `T^2` test for 1 sample: p-value and confidence interval ----------------------------------------------------------------------------------- .. method:: ctx.hotelling_1sample_test(self, ctx, mean, mean0, sd, n, alpha=0.05) where ``ctx`` is ``dec``, ``mpm``, or ``gmp``. Returns the results of tests for a CR or RB Anova. Hotelling's `T^2` test for 1 sample: power and sample size ---------------------------------------------------------------------------------- .. method:: ctx.hotelling_1sample_power(self, ctx, mean, mean0, sd, n, alpha=0.05) where ``ctx`` is ``dec``, ``mpm``, or ``gmp``. Returns the results of tests for a CR or RB Anova. Hotelling's `T^2` test for 2 independent samples: p-value and confidence interval ------------------------------------------------------------------------------------------ .. method:: ctx.hotelling_2isamples_test(self, ctx, mean, mean0, sd, n, alpha=0.05) where ``ctx`` is ``dec``, ``mpm``, or ``gmp``. Returns the results of tests for a CR or RB Anova. Hotelling's `T^2` test for 2 independent samples: power and sample size ---------------------------------------------------------------------------------------- .. method:: ctx.hotelling_2isamples_power(self, ctx, mean, mean0, sd, n, alpha=0.05) where ``ctx`` is ``dec``, ``mpm``, or ``gmp``. Returns the results of tests for a CR or RB Anova. |newpage| Overview: 4 test criteria for 3 hypotheses ------------------------------------------------------------------------------- Test explaining, based on Tretter and Bortz. .. _rst_mpm_manova: MANOVA: Wilks `\Lambda`, Pillai's `V`, Hotelling's `T^2`, Roy's largest root `\theta` ------------------------------------------------------------------------------------------ .. method:: ctx.four_tests_glm_test(x, p, m, n, cdf=True, method='default') where ``ctx`` is ``dec``, ``mpm``, or ``gmp``. The Lawley-Hotelling generalized `T_0^2` and Pillai's `V` statistic, defined respectively by .. math:: T_0^2 = n \text{tr} (AB^{-1}), \quad V = n \text{tr} (A(A+B)^{-1}), have been suggested as alternatives to Wilk's criterion for testing multivariate linear hypotheses. Here `A` and `B` are independent `p \times p` Wishart matrices on `q` and `n` degrees of freedom respectively. .. _rst_mpm_cancorr: Canonical correlation: Wilks `\Lambda`, Pillai's `V`, Hotelling's `T^2`, Roy's largest root `\theta` ------------------------------------------------------------------------------------------------------- .. method:: ctx.four_tests_ind_test(x, p, m, n, cdf=True, method='default') where ``ctx`` is ``dec``, ``mpm``, or ``gmp``. Some Text Power estimates of 4 tests in MANOVA ------------------------------------------------------------------------------- .. method:: ctx.four_tests_glm_power(x, p, m, n, Omega, cdf=True, method='default') where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Returns `\text{cdf}_X(x)`, the cumulative distribution function, for ``cdf=True``, or `\text{sf}_X(x)`, the survival function, for ``cdf=False``, of a random variable `X`, following a noncentral Wilks’ `\Lambda` distribution under the GLM alternative,, with `p \ge 1` predictor variables, error degress of freedom `m \ge 1` and `n \ge 1`, noncentrality parameter `\Omega` with diagonal entries `\omega_{jj} \in (0,\infty)` and the support interval `(0,1)`. This approach is discussed in detail in chapter ... There are other ways of calculating `\text{pdf}_X(x)` as well. If ``method`` is not specified, the algorithm is chosen automatically. For ``fpm.`` the default method is to call the function provided by Boost. Otherwise, the default is verified integration. ``method='finite_series'``: the finite series described in ... is used. ``method='infinite_series'``: the infinite series described in ... is used. ``method='edgeworth'``: the edgeworth expansion described in ... is used. ``method='lugannini_rice'``: the Lugannini-Rice saddlepoint approximation described in ... is used. An example (CDF): .. code-block:: pycon >>> from mpfunlab import dec, mpm, ipm, fpm, gmp, apm >>> mpm.dps = 40; x = '0.3'; p = '3'; m = '4'; n = '20'; O = [3.6, 1.2, 0.5] >>> dx = dec.four_tests_glm_cdf(x,p,m,n,O); mx = mpm.four_tests_glm_cdf(x,p,m,n,O) >>> ix = ipm.four_tests_glm_cdf(x,p,m,n,O); fx = fpm.four_tests_glm_cdf(x,p,m,n,O) >>> gx = gmp.four_tests_glm_cdf(x,p,m,n,O); ax = apm.four_tests_glm_cdf(x,p,m,n,O) >>> mpm.show([dx, mx, ix, fx, gx, ax]) dec: 9.727307040581953720491613246746146674676E-1 mpm: 9.727307040581953720491613246746146674676e-1 ipm: 9.727307040581953720491613246746146674676e-1 (5.901e-40%) fpm: 9.72730704058195E-01 gmp: 9.727307040581953720491613246746146674676E-01 ipm: 9.727307040581953720491613246746146674676e-1 (5.901e-40%) Power estimates of 4 tests in canonical correlation (Type I) ------------------------------------------------------------------------------- .. method:: ctx.four_tests_ind_power(x, p1, p2, n, Rho2, cdf=True, method='default') where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Returns `\text{cdf}_X(x)`, the cumulative distribution function, for ``cdf=True``, or `\text{sf}_X(x)`, the survival function, for ``cdf=False``, of a random variable `X`, following a noncentral Wilks’ `\Lambda` distribution under the independence alternative, with `p_1 \ge 1` and `p_2 \ge 1` groups of variables, error degress of freedom `n \ge 1`, noncentrality parameter `P^2` with diagonal entries `\rho^2_{jj} \in (0,1)` and the support interval `(0,1)`. This approach is discussed in detail in chapter ... There are other ways of calculating `\text{pdf}_X(x)` as well. If ``method`` is not specified, the algorithm is chosen automatically. For ``fpm.`` the default method is to call the function provided by Boost. Otherwise, the default is verified integration. ``method='finite_series'``: the finite series described in ... is used. ``method='infinite_series'``: the infinite series described in ... is used. ``method='edgeworth'``: the edgeworth expansion described in ... is used. ``method='lugannini_rice'``: the Lugannini-Rice saddlepoint approximation described in ... is used. An example (CDF): .. code-block:: pycon >>> from mpfunlab import dec, mpm, ipm, fpm, gmp, apm >>> mpm.dps = 40; x = '0.3'; p1 = '3'; p2 = '4'; n = '20'; P = [3.6, 1.2, 0.5] >>> dx = dec.four_tests_ind_cdf(x,p1,p2,n,P); mx = mpm.four_tests_ind_cdf(x,p1,p2,n,P) >>> ix = ipm.four_tests_ind_cdf(x,p1,p2,n,P); fx = fpm.four_tests_ind_cdf(x,p1,p2,n,P) >>> gx = gmp.four_tests_ind_cdf(x,p1,p2,n,P); ax = apm.four_tests_ind_cdf(x,p1,p2,n,P) >>> mpm.show([dx, mx, ix, fx, gx, ax]) dec: 9.727307040581953720491613246746146674676E-1 mpm: 9.727307040581953720491613246746146674676e-1 ipm: 9.727307040581953720491613246746146674676e-1 (5.901e-40%) fpm: 9.72730704058195E-01 gmp: 9.727307040581953720491613246746146674676E-01 ipm: 9.727307040581953720491613246746146674676e-1 (5.901e-40%)