# Author: Raphael Vallat import warnings from collections.abc import Iterable import numpy as np import pandas as pd from scipy.stats import f import pandas_flavor as pf from pingouin import ( _check_dataframe, remove_na, _flatten_list, bayesfactor_ttest, epsilon, sphericity, _postprocess_dataframe, ) __all__ = ["ttest", "rm_anova", "anova", "welch_anova", "mixed_anova", "ancova"] def ttest(x, y, paired=False, alternative="two-sided", correction="auto", r=0.707, confidence=0.95): """T-test. Parameters ---------- x : array_like First set of observations. y : array_like or float Second set of observations. If ``y`` is a single value, a one-sample T-test is computed against that value (= "mu" in the t.test R function). paired : boolean Specify whether the two observations are related (i.e. repeated measures) or independent. alternative : string Defines the alternative hypothesis, or tail of the test. Must be one of "two-sided" (default), "greater" or "less". Both "greater" and "less" return one-sided p-values. "greater" tests against the alternative hypothesis that the mean of ``x`` is greater than the mean of ``y``. correction : string or boolean For unpaired two sample T-tests, specify whether or not to correct for unequal variances using Welch separate variances T-test. If 'auto', it will automatically uses Welch T-test when the sample sizes are unequal, as recommended by Zimmerman 2004. r : float Cauchy scale factor for computing the Bayes Factor. Smaller values of r (e.g. 0.5), may be appropriate when small effect sizes are expected a priori; larger values of r are appropriate when large effect sizes are expected (Rouder et al 2009). The default is 0.707 (= :math:`\\sqrt{2} / 2`). confidence : float Confidence level for the confidence intervals (0.95 = 95%) .. versionadded:: 0.3.9 Returns ------- stats : :py:class:`pandas.DataFrame` * ``'T'``: T-value * ``'dof'``: degrees of freedom * ``'alternative'``: alternative of the test * ``'p-val'``: p-value * ``'CI95%'``: confidence intervals of the difference in means * ``'cohen-d'``: Cohen d effect size * ``'BF10'``: Bayes Factor of the alternative hypothesis * ``'power'``: achieved power of the test ( = 1 - type II error) See also -------- mwu, wilcoxon, anova, rm_anova, pairwise_tests, compute_effsize Notes ----- Missing values are automatically removed from the data. If ``x`` and ``y`` are paired, the entire row is removed (= listwise deletion). The **T-value for unpaired samples** is defined as: .. math:: t = \\frac{\\overline{x} - \\overline{y}} {\\sqrt{\\frac{s^{2}_{x}}{n_{x}} + \\frac{s^{2}_{y}}{n_{y}}}} where :math:`\\overline{x}` and :math:`\\overline{y}` are the sample means, :math:`n_{x}` and :math:`n_{y}` are the sample sizes, and :math:`s^{2}_{x}` and :math:`s^{2}_{y}` are the sample variances. The degrees of freedom :math:`v` are :math:`n_x + n_y - 2` when the sample sizes are equal. When the sample sizes are unequal or when :code:`correction=True`, the Welch–Satterthwaite equation is used to approximate the adjusted degrees of freedom: .. math:: v = \\frac{(\\frac{s^{2}_{x}}{n_{x}} + \\frac{s^{2}_{y}}{n_{y}})^{2}} {\\frac{(\\frac{s^{2}_{x}}{n_{x}})^{2}}{(n_{x}-1)} + \\frac{(\\frac{s^{2}_{y}}{n_{y}})^{2}}{(n_{y}-1)}} The p-value is then calculated using a T distribution with :math:`v` degrees of freedom. The T-value for **paired samples** is defined by: .. math:: t = \\frac{\\overline{x}_d}{s_{\\overline{x}}} where .. math:: s_{\\overline{x}} = \\frac{s_d}{\\sqrt n} where :math:`\\overline{x}_d` is the sample mean of the differences between the two paired samples, :math:`n` is the number of observations (sample size), :math:`s_d` is the sample standard deviation of the differences and :math:`s_{\\overline{x}}` is the estimated standard error of the mean of the differences. The p-value is then calculated using a T-distribution with :math:`n-1` degrees of freedom. The scaled Jeffrey-Zellner-Siow (JZS) Bayes Factor is approximated using the :py:func:`pingouin.bayesfactor_ttest` function. Results have been tested against JASP and the `t.test` R function. References ---------- * https://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm * Delacre, M., Lakens, D., & Leys, C. (2017). Why psychologists should by default use Welch’s t-test instead of Student’s t-test. International Review of Social Psychology, 30(1). * Zimmerman, D. W. (2004). A note on preliminary tests of equality of variances. British Journal of Mathematical and Statistical Psychology, 57(1), 173-181. * Rouder, J.N., Speckman, P.L., Sun, D., Morey, R.D., Iverson, G., 2009. Bayesian t tests for accepting and rejecting the null hypothesis. Psychon. Bull. Rev. 16, 225–237. https://doi.org/10.3758/PBR.16.2.225 Examples -------- 1. One-sample T-test. >>> from pingouin import ttest >>> x = [5.5, 2.4, 6.8, 9.6, 4.2] >>> ttest(x, 4).round(2) T dof alternative p-val CI95% cohen-d BF10 power T-test 1.4 4 two-sided 0.23 [2.32, 9.08] 0.62 0.766 0.19 2. One sided paired T-test. >>> pre = [5.5, 2.4, 6.8, 9.6, 4.2] >>> post = [6.4, 3.4, 6.4, 11., 4.8] >>> ttest(pre, post, paired=True, alternative='less').round(2) T dof alternative p-val CI95% cohen-d BF10 power T-test -2.31 4 less 0.04 [-inf, -0.05] 0.25 3.122 0.12 Now testing the opposite alternative hypothesis >>> ttest(pre, post, paired=True, alternative='greater').round(2) T dof alternative p-val CI95% cohen-d BF10 power T-test -2.31 4 greater 0.96 [-1.35, inf] 0.25 0.32 0.02 3. Paired T-test with missing values. >>> import numpy as np >>> pre = [5.5, 2.4, np.nan, 9.6, 4.2] >>> post = [6.4, 3.4, 6.4, 11., 4.8] >>> ttest(pre, post, paired=True).round(3) T dof alternative p-val CI95% cohen-d BF10 power T-test -5.902 3 two-sided 0.01 [-1.5, -0.45] 0.306 7.169 0.073 Compare with SciPy >>> from scipy.stats import ttest_rel >>> np.round(ttest_rel(pre, post, nan_policy="omit"), 3) array([-5.902, 0.01 ]) 4. Independent two-sample T-test with equal sample size. >>> np.random.seed(123) >>> x = np.random.normal(loc=7, size=20) >>> y = np.random.normal(loc=4, size=20) >>> ttest(x, y) T dof alternative p-val CI95% cohen-d BF10 power T-test 9.106452 38 two-sided 4.306971e-11 [2.64, 4.15] 2.879713 1.366e+08 1.0 5. Independent two-sample T-test with unequal sample size. A Welch's T-test is used. >>> np.random.seed(123) >>> y = np.random.normal(loc=6.5, size=15) >>> ttest(x, y) T dof alternative p-val CI95% cohen-d BF10 power T-test 1.996537 31.567592 two-sided 0.054561 [-0.02, 1.65] 0.673518 1.469 0.481867 6. However, the Welch's correction can be disabled: >>> ttest(x, y, correction=False) T dof alternative p-val CI95% cohen-d BF10 power T-test 1.971859 33 two-sided 0.057056 [-0.03, 1.66] 0.673518 1.418 0.481867 Compare with SciPy >>> from scipy.stats import ttest_ind >>> np.round(ttest_ind(x, y, equal_var=True), 6) # T value and p-value array([1.971859, 0.057056]) """ from scipy.stats import t, ttest_rel, ttest_ind, ttest_1samp try: # pragma: no cover from scipy.stats._stats_py import _unequal_var_ttest_denom, _equal_var_ttest_denom except ImportError: # pragma: no cover # Fallback for scipy<1.8.0 from scipy.stats.stats import _unequal_var_ttest_denom, _equal_var_ttest_denom from pingouin import power_ttest, power_ttest2n, compute_effsize # Check arguments assert alternative in [ "two-sided", "greater", "less", ], "Alternative must be one of 'two-sided' (default), 'greater' or 'less'." assert 0 < confidence < 1, "confidence must be between 0 and 1." x = np.asarray(x) y = np.asarray(y) if x.size != y.size and paired: warnings.warn("x and y have unequal sizes. Switching to paired == False. Check your data.") paired = False # Remove rows with missing values x, y = remove_na(x, y, paired=paired) nx, ny = x.size, y.size if ny == 1: # Case one sample T-test tval, pval = ttest_1samp(x, y, alternative=alternative) # Some versions of scipy return an array for the t-value if isinstance(tval, Iterable): tval = tval[0] dof = nx - 1 se = np.sqrt(x.var(ddof=1) / nx) if ny > 1 and paired is True: # Case paired two samples T-test # Do not compute if two arrays are identical (avoid SciPy warning) if np.array_equal(x, y): warnings.warn("x and y are equals. Cannot compute T or p-value.") tval, pval = np.nan, np.nan else: tval, pval = ttest_rel(x, y, alternative=alternative) dof = nx - 1 se = np.sqrt(np.var(x - y, ddof=1) / nx) bf = bayesfactor_ttest(tval, nx, ny, paired=True, r=r) elif ny > 1 and paired is False: dof = nx + ny - 2 vx, vy = x.var(ddof=1), y.var(ddof=1) # Case unpaired two samples T-test if correction is True or (correction == "auto" and nx != ny): # Use the Welch separate variance T-test tval, pval = ttest_ind(x, y, equal_var=False, alternative=alternative) # Compute sample standard deviation # dof are approximated using Welch–Satterthwaite equation dof, se = _unequal_var_ttest_denom(vx, nx, vy, ny) else: tval, pval = ttest_ind(x, y, equal_var=True, alternative=alternative) _, se = _equal_var_ttest_denom(vx, nx, vy, ny) # Effect size d = compute_effsize(x, y, paired=paired, eftype="cohen") # Confidence interval for the (difference in) means # Compare to the t.test r function if alternative == "two-sided": alpha = 1 - confidence conf = 1 - alpha / 2 # 0.975 else: conf = confidence tcrit = t.ppf(conf, dof) ci = np.array([tval - tcrit, tval + tcrit]) * se if ny == 1: ci += y if alternative == "greater": ci[1] = np.inf elif alternative == "less": ci[0] = -np.inf # Rename CI ci_name = "CI%.0f%%" % (100 * confidence) # Achieved power if ny == 1: # One-sample power = power_ttest( d=d, n=nx, power=None, alpha=0.05, contrast="one-sample", alternative=alternative ) if ny > 1 and paired is True: # Paired two-sample power = power_ttest( d=d, n=nx, power=None, alpha=0.05, contrast="paired", alternative=alternative ) elif ny > 1 and paired is False: # Independent two-samples if nx == ny: # Equal sample sizes power = power_ttest( d=d, n=nx, power=None, alpha=0.05, contrast="two-samples", alternative=alternative ) else: # Unequal sample sizes power = power_ttest2n(nx, ny, d=d, power=None, alpha=0.05, alternative=alternative) # Bayes factor bf = bayesfactor_ttest(tval, nx, ny, paired=paired, alternative=alternative, r=r) # Create output dictionnary stats = { "dof": dof, "T": tval, "p-val": pval, "alternative": alternative, "cohen-d": abs(d), ci_name: [ci], "power": power, "BF10": bf, } # Convert to dataframe col_order = ["T", "dof", "alternative", "p-val", ci_name, "cohen-d", "BF10", "power"] stats = pd.DataFrame(stats, columns=col_order, index=["T-test"]) return _postprocess_dataframe(stats) @pf.register_dataframe_method def rm_anova( data=None, dv=None, within=None, subject=None, correction="auto", detailed=False, effsize="ng2" ): """One-way and two-way repeated measures ANOVA. Parameters ---------- data : :py:class:`pandas.DataFrame` DataFrame. Note that this function can also directly be used as a :py:class:`pandas.DataFrame` method, in which case this argument is no longer needed. Both wide and long-format dataframe are supported for one-way repeated measures ANOVA. However, ``data`` must be in long format for two-way repeated measures. dv : string Name of column containing the dependent variable (only required if ``data`` is in long format). within : string or list of string Name of column containing the within factor (only required if ``data`` is in long format). If ``within`` is a single string, then compute a one-way repeated measures ANOVA, if ``within`` is a list with two strings, compute a two-way repeated measures ANOVA. subject : string Name of column containing the subject identifier (only required if ``data`` is in long format). correction : string or boolean If True, also return the Greenhouse-Geisser corrected p-value. The default for one-way design is to compute Mauchly's test of sphericity to determine whether the p-values needs to be corrected (see :py:func:`pingouin.sphericity`). The default for two-way design is to return both the uncorrected and Greenhouse-Geisser corrected p-values. Note that sphericity test for two-way design are not currently implemented in Pingouin. detailed : boolean If True, return a full ANOVA table. effsize : string Effect size. Must be one of 'np2' (partial eta-squared), 'n2' (eta-squared) or 'ng2'(generalized eta-squared, default). Note that for one-way repeated measure ANOVA, eta-squared is the same as the generalized eta-squared. Returns ------- aov : :py:class:`pandas.DataFrame` ANOVA summary: * ``'Source'``: Name of the within-group factor * ``'ddof1'``: Degrees of freedom (numerator) * ``'ddof2'``: Degrees of freedom (denominator) * ``'F'``: F-value * ``'p-unc'``: Uncorrected p-value * ``'ng2'``: Generalized eta-square effect size * ``'eps'``: Greenhouse-Geisser epsilon factor (= index of sphericity) * ``'p-GG-corr'``: Greenhouse-Geisser corrected p-value * ``'W-spher'``: Sphericity test statistic * ``'p-spher'``: p-value of the sphericity test * ``'sphericity'``: sphericity of the data (boolean) See Also -------- anova : One-way and N-way ANOVA mixed_anova : Two way mixed ANOVA friedman : Non-parametric one-way repeated measures ANOVA Notes ----- Data can be in wide or long format for one-way repeated measures ANOVA but *must* be in long format for two-way repeated measures ANOVA. In one-way repeated-measures ANOVA, the total variance (sums of squares) is divided into three components .. math:: SS_{\\text{total}} = SS_{\\text{effect}} + (SS_{\\text{subjects}} + SS_{\\text{error}}) with .. math:: SS_{\\text{total}} = \\sum_i^r \\sum_j^n (Y_{ij} - \\overline{Y})^2 SS_{\\text{effect}} = \\sum_i^r n_i(\\overline{Y_i} - \\overline{Y})^2 SS_{\\text{subjects}} = r\\sum (\\overline{Y}_s - \\overline{Y})^2 SS_{\\text{error}} = SS_{\\text{total}} - SS_{\\text{effect}} - SS_{\\text{subjects}} where :math:`i=1,...,r; j=1,...,n_i`, :math:`r` is the number of conditions, :math:`n_i` the number of observations for each condition, :math:`\\overline{Y}` the grand mean of the data, :math:`\\overline{Y_i}` the mean of the :math:`i^{th}` condition and :math:`\\overline{Y}_{subj}` the mean of the :math:`s^{th}` subject. The F-statistics is then defined as: .. math:: F^* = \\frac{MS_{\\text{effect}}}{MS_{\\text{error}}} = \\frac{\\frac{SS_{\\text{effect}}} {r-1}}{\\frac{SS_{\\text{error}}}{(n - 1)(r - 1)}} and the p-value can be calculated using a F-distribution with :math:`v_{\\text{effect}} = r - 1` and :math:`v_{\\text{error}} = (n - 1)(r - 1)` degrees of freedom. The default effect size reported in Pingouin is the generalized eta-squared, which is equivalent to eta-squared for one-way repeated measures ANOVA. .. math:: \\eta_g^2 = \\frac{SS_{\\text{effect}}}{SS_{\\text{total}}} The partial eta-squared is defined as: .. math:: \\eta_p^2 = \\frac{SS_{\\text{effect}}}{SS_{\\text{effect}} + SS_{\\text{error}}} Missing values are automatically removed using a strict listwise approach (= complete-case analysis). In other words, any subject with one or more missing value(s) is completely removed from the dataframe prior to running the test. This could drastically decrease the power of the ANOVA if many missing values are present. In that case, we strongly recommend using linear mixed effect modelling, which can handle missing values in repeated measures. .. warning:: The epsilon adjustement factor of the interaction in two-way repeated measures ANOVA where both factors have more than two levels slightly differs than from R and JASP. Please always make sure to double-check your results with another software. .. warning:: Sphericity tests for the interaction term of a two-way repeated measures ANOVA are not currently supported in Pingouin. Instead, please refer to the Greenhouse-Geisser epsilon value (a value close to 1 indicates that sphericity is met.) For more details, see :py:func:`pingouin.sphericity`. Examples -------- 1. One-way repeated measures ANOVA using a wide-format dataset >>> import pingouin as pg >>> data = pg.read_dataset('rm_anova_wide') >>> pg.rm_anova(data) Source ddof1 ddof2 F p-unc ng2 eps 0 Within 3 24 5.200652 0.006557 0.346392 0.694329 2. One-way repeated-measures ANOVA using a long-format dataset. We're also specifying two additional options here: ``detailed=True`` means that we'll get a more detailed ANOVA table, and ``effsize='np2'`` means that we want to get the partial eta-squared effect size instead of the default (generalized) eta-squared. >>> df = pg.read_dataset('rm_anova') >>> aov = pg.rm_anova(dv='DesireToKill', within='Disgustingness', ... subject='Subject', data=df, detailed=True, effsize="np2") >>> aov.round(3) Source SS DF MS F p-unc np2 eps 0 Disgustingness 27.485 1 27.485 12.044 0.001 0.116 1.0 1 Error 209.952 92 2.282 NaN NaN NaN NaN 3. Two-way repeated-measures ANOVA >>> aov = pg.rm_anova(dv='DesireToKill', within=['Disgustingness', 'Frighteningness'], ... subject='Subject', data=df) 4. As a :py:class:`pandas.DataFrame` method >>> df.rm_anova(dv='DesireToKill', within='Disgustingness', subject='Subject', detailed=False) Source ddof1 ddof2 F p-unc ng2 eps 0 Disgustingness 1 92 12.043878 0.000793 0.025784 1.0 """ assert effsize in ["n2", "np2", "ng2"], "effsize must be n2, np2 or ng2." if isinstance(within, list): assert len(within) > 0, "Within cannot be empty." if len(within) == 1: within = within[0] elif len(within) == 2: return rm_anova2(dv=dv, within=within, data=data, subject=subject, effsize=effsize) else: raise ValueError("Repeated measures ANOVA with three or more factors is not supported.") # Convert from wide to long-format, if needed if all([v is None for v in [dv, within, subject]]): assert isinstance(data, pd.DataFrame) data = data._get_numeric_data().dropna() # Listwise deletion of missing values assert data.shape[0] > 2, "Data must have at least 3 non-missing rows." assert data.shape[1] > 1, "Data must contain at least two columns." data["Subj"] = np.arange(data.shape[0]) data = data.melt(id_vars="Subj", var_name="Within", value_name="DV") subject, within, dv = "Subj", "Within", "DV" # Check dataframe data = _check_dataframe(dv=dv, within=within, data=data, subject=subject, effects="within") assert not data[within].isnull().any(), "Cannot have missing values in `within`." assert not data[subject].isnull().any(), "Cannot have missing values in `subject`." # Pivot and melt the table. This has several effects: # 1) Force missing values to be explicit (a NaN cell is created) # 2) Automatic collapsing to the mean if multiple within factors are present # 3) If using dropna, remove rows with missing values (listwise deletion). # The latter is the same behavior as JASP (= strict complete-case analysis). data_piv = data.pivot_table(index=subject, columns=within, values=dv, observed=True) data_piv = data_piv.dropna() data = data_piv.melt(ignore_index=False, value_name=dv).reset_index() # Groupby # I think that observed=True is actually not needed here since we have already used # `observed=True` in pivot_table. grp_with = data.groupby(within, observed=True, group_keys=False)[dv] rm = list(data[within].unique()) n_rm = len(rm) n_obs = int(grp_with.count().max()) grandmean = data[dv].mean(numeric_only=True) # Calculate sums of squares ss_with = ((grp_with.mean(numeric_only=True) - grandmean) ** 2 * grp_with.count()).sum() ss_resall = grp_with.apply(lambda x: (x - x.mean()) ** 2).sum() # sstotal = sstime + ss_resall = sstime + (sssubj + sserror) # ss_total = ((data[dv] - grandmean)**2).sum() # We can further divide the residuals into a within and between component: grp_subj = data.groupby(subject, observed=True)[dv] ss_resbetw = n_rm * np.sum((grp_subj.mean(numeric_only=True) - grandmean) ** 2) ss_reswith = ss_resall - ss_resbetw # Calculate degrees of freedom ddof1 = n_rm - 1 ddof2 = ddof1 * (n_obs - 1) # Calculate MS, F and p-values ms_with = ss_with / ddof1 ms_reswith = ss_reswith / ddof2 fval = ms_with / ms_reswith p_unc = f(ddof1, ddof2).sf(fval) # Calculating effect sizes (see Bakeman 2005; Lakens 2013) # https://github.com/raphaelvallat/pingouin/issues/251 if effsize == "np2": # Partial eta-squared ef = ss_with / (ss_with + ss_reswith) else: # (Generalized) eta-squared, ng2 == n2 ef = ss_with / (ss_with + ss_resall) # Compute sphericity using Mauchly test, on the wide-format dataframe # Sphericity assumption only applies if there are more than 2 levels if correction == "auto" or (correction is True and n_rm >= 3): spher, W_spher, _, _, p_spher = sphericity(data_piv, alpha=0.05) if correction == "auto": correction = True if not spher else False else: correction = False # Compute epsilon adjustement factor eps = epsilon(data_piv, correction="gg") # If required, apply Greenhouse-Geisser correction for sphericity if correction: corr_ddof1, corr_ddof2 = (np.maximum(d * eps, 1.0) for d in (ddof1, ddof2)) p_corr = f(corr_ddof1, corr_ddof2).sf(fval) # Create output dataframe if not detailed: aov = pd.DataFrame( { "Source": within, "ddof1": ddof1, "ddof2": ddof2, "F": fval, "p-unc": p_unc, effsize: ef, "eps": eps, }, index=[0], ) if correction: aov["p-GG-corr"] = p_corr aov["W-spher"] = W_spher aov["p-spher"] = p_spher aov["sphericity"] = spher col_order = [ "Source", "ddof1", "ddof2", "F", "p-unc", "p-GG-corr", effsize, "eps", "sphericity", "W-spher", "p-spher", ] else: aov = pd.DataFrame( { "Source": [within, "Error"], "SS": [ss_with, ss_reswith], "DF": [ddof1, ddof2], "MS": [ms_with, ms_reswith], "F": [fval, np.nan], "p-unc": [p_unc, np.nan], effsize: [ef, np.nan], "eps": [eps, np.nan], } ) if correction: aov["p-GG-corr"] = [p_corr, np.nan] aov["W-spher"] = [W_spher, np.nan] aov["p-spher"] = [p_spher, np.nan] aov["sphericity"] = [spher, np.nan] col_order = [ "Source", "SS", "DF", "MS", "F", "p-unc", "p-GG-corr", effsize, "eps", "sphericity", "W-spher", "p-spher", ] aov = aov.reindex(columns=col_order) aov.dropna(how="all", axis=1, inplace=True) return _postprocess_dataframe(aov) def rm_anova2(data=None, dv=None, within=None, subject=None, effsize="ng2"): """Two-way repeated measures ANOVA. This is an internal function. The main call to this function should be done by the :py:func:`pingouin.rm_anova` function. """ a, b = within # Validate the dataframe data = _check_dataframe(dv=dv, within=within, data=data, subject=subject, effects="within") assert not data[a].isnull().any(), "Cannot have missing values in %s" % a assert not data[b].isnull().any(), "Cannot have missing values in %s" % b assert not data[subject].isnull().any(), "Cannot have missing values in %s" % subject # Pivot and melt the table. This has several effects: # 1) Force missing values to be explicit (a NaN cell is created) # 2) Automatic collapsing to the mean if multiple within factors are present # 3) If using dropna, remove rows with missing values (listwise deletion). # The latter is the same behavior as JASP (= strict complete-case analysis). data_piv = data.pivot_table(index=subject, columns=within, values=dv, observed=True) data_piv = data_piv.dropna() data = data_piv.melt(ignore_index=False, value_name=dv).reset_index() # Group sizes and grandmean n_a = data[a].nunique() n_b = data[b].nunique() n_s = data[subject].nunique() mu = data[dv].mean(numeric_only=True) # Groupby means # I think that observed=True is actually not needed here since we have already used # `observed=True` in pivot_table. grp_s = data.groupby(subject, observed=True)[dv].mean(numeric_only=True) grp_a = data.groupby([a], observed=True)[dv].mean(numeric_only=True) grp_b = data.groupby([b], observed=True)[dv].mean(numeric_only=True) grp_ab = data.groupby([a, b], observed=True)[dv].mean(numeric_only=True) grp_as = data.groupby([a, subject], observed=True)[dv].mean(numeric_only=True) grp_bs = data.groupby([b, subject], observed=True)[dv].mean(numeric_only=True) # Sums of squares ss_tot = np.sum((data[dv] - mu) ** 2) ss_s = (n_a * n_b) * np.sum((grp_s - mu) ** 2) ss_a = (n_b * n_s) * np.sum((grp_a - mu) ** 2) ss_b = (n_a * n_s) * np.sum((grp_b - mu) ** 2) ss_ab_er = n_s * np.sum((grp_ab - mu) ** 2) ss_ab = ss_ab_er - ss_a - ss_b ss_as_er = n_b * np.sum((grp_as - mu) ** 2) ss_as = ss_as_er - ss_s - ss_a ss_bs_er = n_a * np.sum((grp_bs - mu) ** 2) ss_bs = ss_bs_er - ss_s - ss_b ss_abs = ss_tot - ss_a - ss_b - ss_s - ss_ab - ss_as - ss_bs # DOF df_a = n_a - 1 df_b = n_b - 1 df_s = n_s - 1 df_ab_er = n_a * n_b - 1 df_ab = df_ab_er - df_a - df_b df_as_er = n_a * n_s - 1 df_as = df_as_er - df_s - df_a df_bs_er = n_b * n_s - 1 df_bs = df_bs_er - df_s - df_b df_tot = n_a * n_b * n_s - 1 df_abs = df_tot - df_a - df_b - df_s - df_ab - df_as - df_bs # Mean squares ms_a = ss_a / df_a ms_b = ss_b / df_b ms_ab = ss_ab / df_ab ms_as = ss_as / df_as ms_bs = ss_bs / df_bs ms_abs = ss_abs / df_abs # F-values f_a = ms_a / ms_as f_b = ms_b / ms_bs f_ab = ms_ab / ms_abs # P-values p_a = f(df_a, df_as).sf(f_a) p_b = f(df_b, df_bs).sf(f_b) p_ab = f(df_ab, df_abs).sf(f_ab) # Effect sizes if effsize == "n2": # ..Eta-squared ef_a = ss_a / ss_tot ef_b = ss_b / ss_tot ef_ab = ss_ab / ss_tot elif effsize == "ng2": # .. Generalized eta-squared (from Bakeman 2005 Table 1) -- default ef_a = ss_a / (ss_a + ss_s + ss_as + ss_bs + ss_abs) ef_b = ss_b / (ss_b + ss_s + ss_as + ss_bs + ss_abs) ef_ab = ss_ab / (ss_ab + ss_s + ss_as + ss_bs + ss_abs) else: # .. Partial eta squared ef_a = (f_a * df_a) / (f_a * df_a + df_as) ef_b = (f_b * df_b) / (f_b * df_b + df_bs) ef_ab = (f_ab * df_ab) / (f_ab * df_ab + df_abs) # Epsilon piv_a = data.pivot_table(index=subject, columns=a, values=dv, observed=True) piv_b = data.pivot_table(index=subject, columns=b, values=dv, observed=True) eps_a = epsilon(piv_a, correction="gg") eps_b = epsilon(piv_b, correction="gg") # Note that the GG epsilon of the interaction slightly differs between # R and Pingouin. An alternative is to use the lower bound, which is # very conservative (same behavior as described on real-statistics.com). eps_ab = epsilon(data_piv, correction="gg") # Greenhouse-Geisser correction df_a_c, df_as_c = (np.maximum(d * eps_a, 1.0) for d in (df_a, df_as)) df_b_c, df_bs_c = (np.maximum(d * eps_b, 1.0) for d in (df_b, df_bs)) df_ab_c, df_abs_c = (np.maximum(d * eps_ab, 1.0) for d in (df_ab, df_abs)) p_a_corr = f(df_a_c, df_as_c).sf(f_a) p_b_corr = f(df_b_c, df_bs_c).sf(f_b) p_ab_corr = f(df_ab_c, df_abs_c).sf(f_ab) # Create dataframe aov = pd.DataFrame( { "Source": [a, b, a + " * " + b], "SS": [ss_a, ss_b, ss_ab], "ddof1": [df_a, df_b, df_ab], "ddof2": [df_as, df_bs, df_abs], "MS": [ms_a, ms_b, ms_ab], "F": [f_a, f_b, f_ab], "p-unc": [p_a, p_b, p_ab], "p-GG-corr": [p_a_corr, p_b_corr, p_ab_corr], effsize: [ef_a, ef_b, ef_ab], "eps": [eps_a, eps_b, eps_ab], } ) return _postprocess_dataframe(aov) @pf.register_dataframe_method def anova(data=None, dv=None, between=None, ss_type=2, detailed=False, effsize="np2"): """One-way and *N*-way ANOVA. Parameters ---------- data : :py:class:`pandas.DataFrame` DataFrame. Note that this function can also directly be used as a Pandas method, in which case this argument is no longer needed. dv : string Name of column in ``data`` containing the dependent variable. between : string or list with *N* elements Name of column(s) in ``data`` containing the between-subject factor(s). If ``between`` is a single string, a one-way ANOVA is computed. If ``between`` is a list with two or more elements, a *N*-way ANOVA is performed. Note that Pingouin will internally call statsmodels to calculate ANOVA with 3 or more factors, or unbalanced two-way ANOVA. ss_type : int Specify how the sums of squares is calculated for *unbalanced* design with 2 or more factors. Can be 1, 2 (default), or 3. This has no impact on one-way design or N-way ANOVA with balanced data. detailed : boolean If True, return a detailed ANOVA table (default True for N-way ANOVA). effsize : str Effect size. Must be 'np2' (partial eta-squared) or 'n2' (eta-squared). Note that for one-way ANOVA partial eta-squared is the same as eta-squared. Returns ------- aov : :py:class:`pandas.DataFrame` ANOVA summary: * ``'Source'``: Factor names * ``'SS'``: Sums of squares * ``'DF'``: Degrees of freedom * ``'MS'``: Mean squares * ``'F'``: F-values * ``'p-unc'``: uncorrected p-values * ``'np2'``: Partial eta-square effect sizes See Also -------- rm_anova : One-way and two-way repeated measures ANOVA mixed_anova : Two way mixed ANOVA welch_anova : One-way Welch ANOVA kruskal : Non-parametric one-way ANOVA Notes ----- The classic ANOVA is very powerful when the groups are normally distributed and have equal variances. However, when the groups have unequal variances, it is best to use the Welch ANOVA (:py:func:`pingouin.welch_anova`) that better controls for type I error (Liu 2015). The homogeneity of variances can be measured with the :py:func:`pingouin.homoscedasticity` function. The main idea of ANOVA is to partition the variance (sums of squares) into several components. For example, in one-way ANOVA: .. math:: SS_{\\text{total}} = SS_{\\text{effect}} + SS_{\\text{error}} SS_{\\text{total}} = \\sum_i \\sum_j (Y_{ij} - \\overline{Y})^2 SS_{\\text{effect}} = \\sum_i n_i (\\overline{Y_i} - \\overline{Y})^2 SS_{\\text{error}} = \\sum_i \\sum_j (Y_{ij} - \\overline{Y}_i)^2 where :math:`i=1,...,r; j=1,...,n_i`, :math:`r` is the number of groups, and :math:`n_i` the number of observations for the :math:`i` th group. The F-statistics is then defined as: .. math:: F^* = \\frac{MS_{\\text{effect}}}{MS_{\\text{error}}} = \\frac{SS_{\\text{effect}} / (r - 1)}{SS_{\\text{error}} / (n_t - r)} and the p-value can be calculated using a F-distribution with :math:`r-1, n_t-1` degrees of freedom. When the groups are balanced and have equal variances, the optimal post-hoc test is the Tukey-HSD test (:py:func:`pingouin.pairwise_tukey`). If the groups have unequal variances, the Games-Howell test is more adequate (:py:func:`pingouin.pairwise_gameshowell`). The default effect size reported in Pingouin is the partial eta-square, which, for one-way ANOVA is the same as eta-square and generalized eta-square. .. math:: \\eta_p^2 = \\frac{SS_{\\text{effect}}}{SS_{\\text{effect}} + SS_{\\text{error}}} Missing values are automatically removed. Results have been tested against R, Matlab and JASP. Examples -------- One-way ANOVA >>> import pingouin as pg >>> df = pg.read_dataset('anova') >>> aov = pg.anova(dv='Pain threshold', between='Hair color', data=df, ... detailed=True) >>> aov.round(3) Source SS DF MS F p-unc np2 0 Hair color 1360.726 3 453.575 6.791 0.004 0.576 1 Within 1001.800 15 66.787 NaN NaN NaN Same but using a standard eta-squared instead of a partial eta-squared effect size. Also note how here we're using the anova function directly as a method (= built-in function) of our pandas dataframe. In that case, we don't have to specify ``data`` anymore. >>> df.anova(dv='Pain threshold', between='Hair color', detailed=False, ... effsize='n2') Source ddof1 ddof2 F p-unc n2 0 Hair color 3 15 6.791407 0.004114 0.575962 Two-way ANOVA with balanced design >>> data = pg.read_dataset('anova2') >>> data.anova(dv="Yield", between=["Blend", "Crop"]).round(3) Source SS DF MS F p-unc np2 0 Blend 2.042 1 2.042 0.004 0.952 0.000 1 Crop 2736.583 2 1368.292 2.525 0.108 0.219 2 Blend * Crop 2360.083 2 1180.042 2.178 0.142 0.195 3 Residual 9753.250 18 541.847 NaN NaN NaN Two-way ANOVA with unbalanced design (requires statsmodels) >>> data = pg.read_dataset('anova2_unbalanced') >>> data.anova(dv="Scores", between=["Diet", "Exercise"], ... effsize="n2").round(3) Source SS DF MS F p-unc n2 0 Diet 390.625 1.0 390.625 7.423 0.034 0.433 1 Exercise 180.625 1.0 180.625 3.432 0.113 0.200 2 Diet * Exercise 15.625 1.0 15.625 0.297 0.605 0.017 3 Residual 315.750 6.0 52.625 NaN NaN NaN Three-way ANOVA, type 3 sums of squares (requires statsmodels) >>> data = pg.read_dataset('anova3') >>> data.anova(dv='Cholesterol', between=['Sex', 'Risk', 'Drug'], ... ss_type=3).round(3) Source SS DF MS F p-unc np2 0 Sex 2.075 1.0 2.075 2.462 0.123 0.049 1 Risk 11.332 1.0 11.332 13.449 0.001 0.219 2 Drug 0.816 2.0 0.408 0.484 0.619 0.020 3 Sex * Risk 0.117 1.0 0.117 0.139 0.711 0.003 4 Sex * Drug 2.564 2.0 1.282 1.522 0.229 0.060 5 Risk * Drug 2.438 2.0 1.219 1.446 0.245 0.057 6 Sex * Risk * Drug 1.844 2.0 0.922 1.094 0.343 0.044 7 Residual 40.445 48.0 0.843 NaN NaN NaN """ assert effsize in ["np2", "n2"], "effsize must be 'np2' or 'n2'." if isinstance(between, list): if len(between) == 0: raise ValueError("between is empty.") elif len(between) == 1: between = between[0] elif len(between) == 2: # Two factors with balanced design = Pingouin implementation # Two factors with unbalanced design = statsmodels return anova2(dv=dv, between=between, data=data, ss_type=ss_type, effsize=effsize) else: # 3 or more factors with (un)-balanced design = statsmodels return anovan(dv=dv, between=between, data=data, ss_type=ss_type, effsize=effsize) # Check data data = _check_dataframe(dv=dv, between=between, data=data, effects="between") # Drop missing values data = data[[dv, between]].dropna() # Reset index (avoid duplicate axis error) data = data.reset_index(drop=True) n_groups = data[between].nunique() N = data[dv].size # Calculate sums of squares grp = data.groupby(between, observed=True, group_keys=False)[dv] # Between effect ssbetween = ( (grp.mean(numeric_only=True) - data[dv].mean(numeric_only=True)) ** 2 * grp.count() ).sum() # Within effect (= error between) # = (grp.var(ddof=0) * grp.count()).sum() sserror = grp.transform(lambda x: (x - x.mean()) ** 2).sum() # In 1-way ANOVA, sstotal = ssbetween + sserror # sstotal = ssbetween + sserror # Calculate DOF, MS, F and p-values ddof1 = n_groups - 1 ddof2 = N - n_groups msbetween = ssbetween / ddof1 mserror = sserror / ddof2 fval = msbetween / mserror p_unc = f(ddof1, ddof2).sf(fval) # Calculating effect sizes (see Bakeman 2005; Lakens 2013) # In one-way ANOVA, partial eta2 = eta2 = generalized eta2 # Similar to (fval * ddof1) / (fval * ddof1 + ddof2) np2 = ssbetween / (ssbetween + sserror) # = ssbetween / sstotal # Omega-squared # o2 = (ddof1 * (msbetween - mserror)) / (sstotal + mserror) # Create output dataframe if not detailed: aov = pd.DataFrame( { "Source": between, "ddof1": ddof1, "ddof2": ddof2, "F": fval, "p-unc": p_unc, effsize: np2, }, index=[0], ) else: aov = pd.DataFrame( { "Source": [between, "Within"], "SS": [ssbetween, sserror], "DF": [ddof1, ddof2], "MS": [msbetween, mserror], "F": [fval, np.nan], "p-unc": [p_unc, np.nan], effsize: [np2, np.nan], } ) aov.dropna(how="all", axis=1, inplace=True) return _postprocess_dataframe(aov) def anova2(data=None, dv=None, between=None, ss_type=2, effsize="np2"): """Two-way balanced ANOVA in pure Python + Pandas. This is an internal function. The main call to this function should be done by the :py:func:`pingouin.anova` function. """ # Validate the dataframe data = _check_dataframe(dv=dv, between=between, data=data, effects="between") assert len(between) == 2, "Must have exactly two between-factors variables" fac1, fac2 = between # Drop missing values data = data[[dv, fac1, fac2]].dropna() assert data.shape[0] >= 5, "Data must have at least 5 non-missing values." # Reset index (avoid duplicate axis error) data = data.reset_index(drop=True) grp_both = data.groupby(between, observed=True, group_keys=False)[dv] if grp_both.count().nunique() == 1: # BALANCED DESIGN aov_fac1 = anova(data=data, dv=dv, between=fac1, detailed=True) aov_fac2 = anova(data=data, dv=dv, between=fac2, detailed=True) ng1, ng2 = data[fac1].nunique(), data[fac2].nunique() # Sums of squares ss_fac1 = aov_fac1.at[0, "SS"] ss_fac2 = aov_fac2.at[0, "SS"] ss_tot = ((data[dv] - data[dv].mean(numeric_only=True)) ** 2).sum() ss_resid = np.sum(grp_both.apply(lambda x: (x - x.mean()) ** 2)) ss_inter = ss_tot - (ss_resid + ss_fac1 + ss_fac2) # Degrees of freedom df_fac1 = aov_fac1.at[0, "DF"] df_fac2 = aov_fac2.at[0, "DF"] df_inter = (ng1 - 1) * (ng2 - 1) df_resid = data[dv].size - (ng1 * ng2) else: # UNBALANCED DESIGN return anovan(dv=dv, between=between, data=data, ss_type=ss_type, effsize=effsize) # Mean squares ms_fac1 = ss_fac1 / df_fac1 ms_fac2 = ss_fac2 / df_fac2 ms_inter = ss_inter / df_inter ms_resid = ss_resid / df_resid # F-values fval_fac1 = ms_fac1 / ms_resid fval_fac2 = ms_fac2 / ms_resid fval_inter = ms_inter / ms_resid # P-values pval_fac1 = f(df_fac1, df_resid).sf(fval_fac1) pval_fac2 = f(df_fac2, df_resid).sf(fval_fac2) pval_inter = f(df_inter, df_resid).sf(fval_inter) # Effect size if effsize == "n2": # Standard eta-square n2_fac1 = ss_fac1 / ss_tot n2_fac2 = ss_fac2 / ss_tot n2_inter = ss_inter / ss_tot all_effsize = [n2_fac1, n2_fac2, n2_inter, np.nan] else: # ..Partial eta-square np2_fac1 = ss_fac1 / (ss_fac1 + ss_resid) np2_fac2 = ss_fac2 / (ss_fac2 + ss_resid) np2_inter = ss_inter / (ss_inter + ss_resid) all_effsize = [np2_fac1, np2_fac2, np2_inter, np.nan] # Create output dataframe aov = pd.DataFrame( { "Source": [fac1, fac2, fac1 + " * " + fac2, "Residual"], "SS": [ss_fac1, ss_fac2, ss_inter, ss_resid], "DF": [df_fac1, df_fac2, df_inter, df_resid], "MS": [ms_fac1, ms_fac2, ms_inter, ms_resid], "F": [fval_fac1, fval_fac2, fval_inter, np.nan], "p-unc": [pval_fac1, pval_fac2, pval_inter, np.nan], effsize: all_effsize, } ) aov.dropna(how="all", axis=1, inplace=True) return _postprocess_dataframe(aov) def anovan(data=None, dv=None, between=None, ss_type=2, effsize="np2"): """N-way ANOVA using statsmodels. This is an internal function. The main call to this function should be done by the :py:func:`pingouin.anova` function. """ # Check that stasmodels is installed from pingouin.utils import _is_statsmodels_installed _is_statsmodels_installed(raise_error=True) from statsmodels.api import stats from statsmodels.formula.api import ols # Validate the dataframe data = _check_dataframe(dv=dv, between=between, data=data, effects="between") all_cols = _flatten_list([dv, between]) bad_chars = [",", "(", ")", ":"] if not all([c not in v for c in bad_chars for v in all_cols]): err_msg = "comma, bracket, and colon are not allowed in column names." raise ValueError(err_msg) # Drop missing values data = data[all_cols].dropna() assert data.shape[0] >= 5, "Data must have at least 5 non-missing values." # Reset index (avoid duplicate axis error) data = data.reset_index(drop=True) # Create R-like formula # https://patsy.readthedocs.io/en/latest/builtins-reference.html # C marks the data as categorical # Q allows to quote variable that do not meet Python variable name rule # e.g. if variable is "weight.in.kg" or "2A" assert dv not in ["C", "Q"], "`dv` must not be 'C' or 'Q'." assert all(fac not in ["C", "Q"] for fac in between), "`between` must not contain 'C' or 'Q'." formula = "Q('%s') ~ " % dv for fac in between: formula += "C(Q('%s'), Sum) * " % fac formula = formula[:-3] # Remove last * and space # Fit using statsmodels lm = ols(formula, data=data).fit() aov = stats.anova_lm(lm, typ=ss_type) # Convert to Pingouin-like dataframe if ss_type == 1: # statsmodels output is not exactly the same when ss_type = 1 aov = aov[["sum_sq", "df", "F", "PR(>F)"]] if ss_type == 3: # Remove intercept row aov = aov.iloc[1:, :] aov = aov.reset_index() aov = aov.rename(columns={"index": "Source", "sum_sq": "SS", "df": "DF", "PR(>F)": "p-unc"}) aov["MS"] = aov["SS"] / aov["DF"] # Effect size if effsize == "n2": # Get standard eta-square for all effects except residuals (last) all_n2 = (aov["SS"] / aov["SS"].sum()).to_numpy() all_n2[-1] = np.nan aov["n2"] = all_n2 else: aov["np2"] = (aov["F"] * aov["DF"]) / (aov["F"] * aov["DF"] + aov.iloc[-1, 2]) def format_source(x): for fac in between: x = x.replace("C(Q('%s'), Sum)" % fac, fac) return x.replace(":", " * ") aov["Source"] = aov["Source"].apply(format_source) # Re-index and round col_order = ["Source", "SS", "DF", "MS", "F", "p-unc", effsize] aov = aov.reindex(columns=col_order) aov.dropna(how="all", axis=1, inplace=True) # Add formula to dataframe aov = _postprocess_dataframe(aov) aov.formula_ = formula return aov @pf.register_dataframe_method def welch_anova(data=None, dv=None, between=None): """One-way Welch ANOVA. Parameters ---------- data : :py:class:`pandas.DataFrame` DataFrame. Note that this function can also directly be used as a Pandas method, in which case this argument is no longer needed. dv : string Name of column containing the dependent variable. between : string Name of column containing the between factor. Returns ------- aov : :py:class:`pandas.DataFrame` ANOVA summary: * ``'Source'``: Factor names * ``'ddof1'``: Numerator degrees of freedom * ``'ddof2'``: Denominator degrees of freedom * ``'F'``: F-values * ``'p-unc'``: uncorrected p-values * ``'np2'``: Partial eta-squared See Also -------- anova : One-way and N-way ANOVA rm_anova : One-way and two-way repeated measures ANOVA mixed_anova : Two way mixed ANOVA kruskal : Non-parametric one-way ANOVA Notes ----- From Wikipedia: *It is named for its creator, Bernard Lewis Welch, and is an adaptation of Student's t-test, and is more reliable when the two samples have unequal variances and/or unequal sample sizes.* The classic ANOVA is very powerful when the groups are normally distributed and have equal variances. However, when the groups have unequal variances, it is best to use the Welch ANOVA that better controls for type I error (Liu 2015). The homogeneity of variances can be measured with the `homoscedasticity` function. The two other assumptions of normality and independance remain. The main idea of Welch ANOVA is to use a weight :math:`w_i` to reduce the effect of unequal variances. This weight is calculated using the sample size :math:`n_i` and variance :math:`s_i^2` of each group :math:`i=1,...,r`: .. math:: w_i = \\frac{n_i}{s_i^2} Using these weights, the adjusted grand mean of the data is: .. math:: \\overline{Y}_{\\text{welch}} = \\frac{\\sum_{i=1}^r w_i\\overline{Y}_i}{\\sum w} where :math:`\\overline{Y}_i` is the mean of the :math:`i` group. The effect sums of squares is defined as: .. math:: SS_{\\text{effect}} = \\sum_{i=1}^r w_i (\\overline{Y}_i - \\overline{Y}_{\\text{welch}})^2 We then need to calculate a term lambda: .. math:: \\Lambda = \\frac{3\\sum_{i=1}^r(\\frac{1}{n_i-1}) (1 - \\frac{w_i}{\\sum w})^2}{r^2 - 1} from which the F-value can be calculated: .. math:: F_{\\text{welch}} = \\frac{SS_{\\text{effect}} / (r-1)} {1 + \\frac{2\\Lambda(r-2)}{3}} and the p-value approximated using a F-distribution with :math:`(r-1, 1 / \\Lambda)` degrees of freedom. When the groups are balanced and have equal variances, the optimal post-hoc test is the Tukey-HSD test (:py:func:`pingouin.pairwise_tukey`). If the groups have unequal variances, the Games-Howell test is more adequate (:py:func:`pingouin.pairwise_gameshowell`). Results have been tested against R. References ---------- .. [1] Liu, Hangcheng. "Comparing Welch's ANOVA, a Kruskal-Wallis test and traditional ANOVA in case of Heterogeneity of Variance." (2015). .. [2] Welch, Bernard Lewis. "On the comparison of several mean values: an alternative approach." Biometrika 38.3/4 (1951): 330-336. Examples -------- 1. One-way Welch ANOVA on the pain threshold dataset. >>> from pingouin import welch_anova, read_dataset >>> df = read_dataset('anova') >>> aov = welch_anova(dv='Pain threshold', between='Hair color', data=df) >>> aov Source ddof1 ddof2 F p-unc np2 0 Hair color 3 8.329841 5.890115 0.018813 0.575962 """ # Check data data = _check_dataframe(dv=dv, between=between, data=data, effects="between") # Reset index (avoid duplicate axis error) data = data.reset_index(drop=True) # Number of groups r = data[between].nunique() ddof1 = r - 1 # Compute weights and ajusted means grp = data.groupby(between, observed=True, group_keys=False)[dv] weights = grp.count() / grp.var(numeric_only=True) adj_grandmean = (weights * grp.mean(numeric_only=True)).sum() / weights.sum() # Sums of squares (regular and adjusted) ss_res = grp.apply(lambda x: (x - x.mean()) ** 2).sum() ss_bet = ( (grp.mean(numeric_only=True) - data[dv].mean(numeric_only=True)) ** 2 * grp.count() ).sum() ss_betadj = np.sum(weights * np.square(grp.mean(numeric_only=True) - adj_grandmean)) ms_betadj = ss_betadj / ddof1 # Calculate lambda, F-value, p-value and np2 lamb = (3 * np.sum((1 / (grp.count() - 1)) * (1 - (weights / weights.sum())) ** 2)) / ( r**2 - 1 ) ddof2 = 1 / lamb fval = ms_betadj / (1 + (2 * lamb * (r - 2)) / 3) pval = f.sf(fval, ddof1, ddof2) np2 = ss_bet / (ss_bet + ss_res) # Create output dataframe aov = pd.DataFrame( { "Source": between, "ddof1": ddof1, "ddof2": ddof2, "F": fval, "p-unc": pval, "np2": np2, }, index=[0], ) return _postprocess_dataframe(aov) @pf.register_dataframe_method def mixed_anova( data=None, dv=None, within=None, subject=None, between=None, correction="auto", effsize="np2" ): """Mixed-design (split-plot) ANOVA. Parameters ---------- data : :py:class:`pandas.DataFrame` DataFrame. Note that this function can also directly be used as a Pandas method, in which case this argument is no longer needed. dv : string Name of column containing the dependent variable. within : string Name of column containing the within-subject factor (repeated measurements). subject : string Name of column containing the between-subject identifier. between : string Name of column containing the between factor. correction : string or boolean If True, return Greenhouse-Geisser corrected p-value. If `'auto'` (default), compute Mauchly's test of sphericity to determine whether the p-values needs to be corrected. effsize : str Effect size. Must be one of 'np2' (partial eta-squared), 'n2' (eta-squared) or 'ng2'(generalized eta-squared). Returns ------- aov : :py:class:`pandas.DataFrame` ANOVA summary: * ``'Source'``: Names of the factor considered * ``'ddof1'``: Degrees of freedom (numerator) * ``'ddof2'``: Degrees of freedom (denominator) * ``'F'``: F-values * ``'p-unc'``: Uncorrected p-values * ``'np2'``: Partial eta-squared effect sizes * ``'eps'``: Greenhouse-Geisser epsilon factor (= index of sphericity) * ``'p-GG-corr'``: Greenhouse-Geisser corrected p-values * ``'W-spher'``: Sphericity test statistic * ``'p-spher'``: p-value of the sphericity test * ``'sphericity'``: sphericity of the data (boolean) See Also -------- anova, rm_anova, pairwise_tests Notes ----- Data are expected to be in long-format (even the repeated measures). If your data is in wide-format, you can use the :py:func:`pandas.melt()` function to convert from wide to long format. Missing values are automatically removed using a strict listwise approach (= complete-case analysis). In other words, any subject with one or more missing value(s) is completely removed from the dataframe prior to running the test. This could drastically decrease the power of the ANOVA if many missing values are present. In that case, we strongly recommend using linear mixed effect modelling, which can handle missing values in repeated measures. .. warning :: If the between-subject groups are unbalanced (= unequal sample sizes), a type II ANOVA will be computed. Note however that SPSS, JAMOVI and JASP by default return a type III ANOVA, which may lead to slightly different results. Examples -------- For more examples, please refer to the `Jupyter notebooks `_ Compute a two-way mixed model ANOVA. >>> from pingouin import mixed_anova, read_dataset >>> df = read_dataset('mixed_anova') >>> aov = mixed_anova(dv='Scores', between='Group', ... within='Time', subject='Subject', data=df) >>> aov.round(3) Source SS DF1 DF2 MS F p-unc np2 eps 0 Group 5.460 1 58 5.460 5.052 0.028 0.080 NaN 1 Time 7.628 2 116 3.814 4.027 0.020 0.065 0.999 2 Interaction 5.167 2 116 2.584 2.728 0.070 0.045 NaN Same but reporting a generalized eta-squared effect size. Notice how we can also apply this function directly as a method of the dataframe, in which case we do not need to specify ``data=df`` anymore. >>> df.mixed_anova(dv='Scores', between='Group', within='Time', ... subject='Subject', effsize="ng2").round(3) Source SS DF1 DF2 MS F p-unc ng2 eps 0 Group 5.460 1 58 5.460 5.052 0.028 0.031 NaN 1 Time 7.628 2 116 3.814 4.027 0.020 0.042 0.999 2 Interaction 5.167 2 116 2.584 2.728 0.070 0.029 NaN """ assert effsize in ["n2", "np2", "ng2"], "effsize must be n2, np2 or ng2." # Check that only a single within and between factor are provided one_is_list = isinstance(within, list) or isinstance(between, list) both_are_str = isinstance(within, (str, int)) and isinstance(between, (str, int)) if one_is_list or not both_are_str: raise ValueError( "within and between factors must both be strings referring to a column in the data. " "Specifying multiple within and between factors is currently not supported. " "For more information, see: https://github.com/raphaelvallat/pingouin/issues/136" ) # Check data data = _check_dataframe( dv=dv, within=within, between=between, data=data, subject=subject, effects="interaction" ) # Pivot and melt the table. This has several effects: # 1) Force missing values to be explicit (a NaN cell is created) # 2) Automatic collapsing to the mean if multiple within factors are present # 3) If using dropna, remove rows with missing values (listwise deletion). # The latter is the same behavior as JASP (= strict complete-case analysis). data_piv = data.pivot_table(index=[subject, between], columns=within, values=dv, observed=True) data_piv = data_piv.dropna() data = data_piv.melt(ignore_index=False, value_name=dv).reset_index() # Check that subject IDs do not overlap between groups: the subject ID # should have a unique range / set of values for each between-subject # group e.g. group1= 1 --> 20 and group2 = 21 --> 40. if not (data.groupby([subject, within], observed=True)[between].nunique() == 1).all(): raise ValueError( "Subject IDs cannot overlap between groups: each " "group in `%s` must have a unique set of " "subject IDs, e.g. group1 = [1, 2, 3, ..., 10] " "and group2 = [11, 12, 13, ..., 20]" % between ) # SUMS OF SQUARES grandmean = data[dv].mean(numeric_only=True) ss_total = ((data[dv] - grandmean) ** 2).sum() # Extract main effects of within and between factors aov_with = rm_anova( dv=dv, within=within, subject=subject, data=data, correction=correction, detailed=True ) aov_betw = anova(dv=dv, between=between, data=data, detailed=True) ss_betw = aov_betw.at[0, "SS"] ss_with = aov_with.at[0, "SS"] # Extract residuals and interactions grp = data.groupby([between, within], observed=True, group_keys=False)[dv] # ssresall = residuals within + residuals between ss_resall = grp.apply(lambda x: (x - x.mean()) ** 2).sum() # Interaction ss_inter = ss_total - (ss_resall + ss_with + ss_betw) ss_reswith = aov_with.at[1, "SS"] - ss_inter ss_resbetw = ss_total - (ss_with + ss_betw + ss_reswith + ss_inter) # DEGREES OF FREEDOM n_obs = data.groupby(within, observed=True)[dv].count().max() df_with = aov_with.at[0, "DF"] df_betw = aov_betw.at[0, "DF"] df_resbetw = n_obs - data.groupby(between, observed=True)[dv].count().count() df_reswith = df_with * df_resbetw df_inter = aov_with.at[0, "DF"] * aov_betw.at[0, "DF"] # MEAN SQUARES ms_betw = aov_betw.at[0, "MS"] ms_with = aov_with.at[0, "MS"] ms_resbetw = ss_resbetw / df_resbetw ms_reswith = ss_reswith / df_reswith ms_inter = ss_inter / df_inter # F VALUES f_betw = ms_betw / ms_resbetw f_with = ms_with / ms_reswith f_inter = ms_inter / ms_reswith # P-values p_betw = f(df_betw, df_resbetw).sf(f_betw) p_with = f(df_with, df_reswith).sf(f_with) p_inter = f(df_inter, df_reswith).sf(f_inter) # Effects sizes (see Bakeman 2005) if effsize == "n2": # Standard eta-squared ef_betw = ss_betw / ss_total ef_with = ss_with / ss_total ef_inter = ss_inter / ss_total elif effsize == "ng2": # Generalized eta-square ef_betw = ss_betw / (ss_betw + ss_resall) ef_with = ss_with / (ss_with + ss_resall) ef_inter = ss_inter / (ss_inter + ss_resall) else: # Partial eta-squared (default) # ef_betw = f_betw * df_betw / (f_betw * df_betw + df_resbetw) # ef_with = f_with * df_with / (f_with * df_with + df_reswith) ef_betw = ss_betw / (ss_betw + ss_resbetw) ef_with = ss_with / (ss_with + ss_reswith) ef_inter = ss_inter / (ss_inter + ss_reswith) # Stats table aov = pd.concat([aov_betw.drop(1), aov_with.drop(1)], axis=0, sort=False, ignore_index=True) # Update values aov.rename(columns={"DF": "DF1"}, inplace=True) aov.at[0, "F"], aov.at[1, "F"] = f_betw, f_with aov.at[0, "p-unc"], aov.at[1, "p-unc"] = p_betw, p_with aov.at[0, effsize], aov.at[1, effsize] = ef_betw, ef_with aov_inter = pd.DataFrame( { "Source": "Interaction", "SS": ss_inter, "DF1": df_inter, "MS": ms_inter, "F": f_inter, "p-unc": p_inter, effsize: ef_inter, }, index=[2], ) aov = pd.concat([aov, aov_inter], axis=0, sort=False, ignore_index=True) aov["DF2"] = [df_resbetw, df_reswith, df_reswith] aov["eps"] = [np.nan, aov_with.at[0, "eps"], np.nan] col_order = [ "Source", "SS", "DF1", "DF2", "MS", "F", "p-unc", "p-GG-corr", effsize, "eps", "sphericity", "W-spher", "p-spher", ] aov = aov.reindex(columns=col_order) aov.dropna(how="all", axis=1, inplace=True) return _postprocess_dataframe(aov) @pf.register_dataframe_method def ancova(data=None, dv=None, between=None, covar=None, effsize="np2"): """ANCOVA with one or more covariate(s). Parameters ---------- data : :py:class:`pandas.DataFrame` DataFrame. Note that this function can also directly be used as a Pandas method, in which case this argument is no longer needed. dv : string Name of column in data with the dependent variable. between : string Name of column in data with the between factor. covar : string or list Name(s) of column(s) in data with the covariate. effsize : str Effect size. Must be 'np2' (partial eta-squared) or 'n2' (eta-squared). Returns ------- aov : :py:class:`pandas.DataFrame` ANCOVA summary: * ``'Source'``: Names of the factor considered * ``'SS'``: Sums of squares * ``'DF'``: Degrees of freedom * ``'F'``: F-values * ``'p-unc'``: Uncorrected p-values * ``'np2'``: Partial eta-squared Notes ----- Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (dv) are equal across levels of a categorical independent variable (between) often called a treatment, while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates or nuisance variables (covar). Pingouin uses :py:class:`statsmodels.regression.linear_model.OLS` to compute the ANCOVA. .. important:: Rows with missing values are automatically removed (listwise deletion). See Also -------- anova : One-way and N-way ANOVA Examples -------- 1. Evaluate the reading scores of students with different teaching method and family income as a covariate. >>> from pingouin import ancova, read_dataset >>> df = read_dataset('ancova') >>> ancova(data=df, dv='Scores', covar='Income', between='Method') Source SS DF F p-unc np2 0 Method 571.029883 3 3.336482 0.031940 0.244077 1 Income 1678.352687 1 29.419438 0.000006 0.486920 2 Residual 1768.522313 31 NaN NaN NaN 2. Evaluate the reading scores of students with different teaching method and family income + BMI as a covariate. >>> ancova(data=df, dv='Scores', covar=['Income', 'BMI'], between='Method', ... effsize="n2") Source SS DF F p-unc n2 0 Method 552.284043 3 3.232550 0.036113 0.141802 1 Income 1573.952434 1 27.637304 0.000011 0.404121 2 BMI 60.013656 1 1.053790 0.312842 0.015409 3 Residual 1708.508657 30 NaN NaN NaN """ # Import from pingouin.utils import _is_statsmodels_installed _is_statsmodels_installed(raise_error=True) from statsmodels.api import stats from statsmodels.formula.api import ols # Safety checks assert effsize in ["np2", "n2"], "effsize must be 'np2' or 'n2'." assert isinstance(data, pd.DataFrame), "data must be a pandas dataframe." assert isinstance(between, str), ( "between must be a string. Pingouin does not support multiple " "between factors. For more details, please see " "https://github.com/raphaelvallat/pingouin/issues/173." ) assert dv in data.columns, "%s is not in data." % dv assert between in data.columns, "%s is not in data." % between assert isinstance(covar, (str, list)), "covar must be a str or a list." if isinstance(covar, str): covar = [covar] for c in covar: assert c in data.columns, "covariate %s is not in data" % c assert data[c].dtype.kind in "bfi", "covariate %s is not numeric" % c # Drop missing values data = data[_flatten_list([dv, between, covar])].dropna() # Fit ANCOVA model # formula = dv ~ 1 + between + covar1 + covar2 + ... assert dv not in ["C", "Q"], "`dv` must not be 'C' or 'Q'." assert between not in ["C", "Q"], "`between` must not be 'C' or 'Q'." assert all(c not in ["C", "Q"] for c in covar), "`covar` must not contain 'C' or 'Q'." formula = f"Q('{dv}') ~ C(Q('{between}'))" for c in covar: formula += " + Q('%s')" % (c) model = ols(formula, data=data).fit() # Create output dataframe aov = stats.anova_lm(model, typ=2).reset_index() aov.rename( columns={"index": "Source", "sum_sq": "SS", "df": "DF", "PR(>F)": "p-unc"}, inplace=True ) aov.at[0, "Source"] = between for i in range(len(covar)): aov.at[i + 1, "Source"] = covar[i] aov["DF"] = aov["DF"].astype(int) # Add effect sizes if effsize == "n2": all_effsize = (aov["SS"] / aov["SS"].sum()).to_numpy() all_effsize[-1] = np.nan else: ss_resid = aov["SS"].iloc[-1] all_effsize = aov["SS"].apply(lambda x: x / (x + ss_resid)).to_numpy() all_effsize[-1] = np.nan aov[effsize] = all_effsize # Add bw as an attribute (for rm_corr function) aov = _postprocess_dataframe(aov) aov.bw_ = model.params.iloc[-1] return aov