Source code for pingouin.parametric

# Author: Raphael Vallat <raphaelvallat9@gmail.com>
import warnings
import numpy as np
import pandas as pd
from scipy.stats import f
import pandas_flavor as pf
from pingouin import (_check_dataframe, remove_rm_na, remove_na, _flatten_list,
_export_table, bayesfactor_ttest, epsilon, sphericity)

__all__ = ["ttest", "rm_anova", "anova", "welch_anova", "mixed_anova",
"ancova"]

[docs]def ttest(x, y, paired=False, tail='two-sided', correction='auto', r=.707): """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. paired : boolean Specify whether the two observations are related (i.e. repeated measures) or independent. tail : string Specify whether the alternative hypothesis is 'two-sided' or 'one-sided'. Can also be 'greater' or 'less' to specify the direction of the test. 'greater' tests the alternative that x has a larger mean than y. If tail is 'one-sided', Pingouin will automatically infer the one-sided alternative hypothesis based on the test statistic. 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). Returns ------- stats : :py:class:pandas.DataFrame T-test summary:: 'T' : T-value 'p-val' : p-value 'dof' : degrees of freedom 'cohen-d' : Cohen d effect size 'CI95%' : 95% confidence intervals of the difference in means 'power' : achieved power of the test ( = 1 - type II error) 'BF10' : Bayes Factor of the alternative hypothesis See also -------- mwu, wilcoxon, anova, rm_anova, pairwise_ttests, 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 **two-sample T-test for unpaired data** 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 tail p-val CI95% cohen-d BF10 power T-test 1.4 4 two-sided 0.23 [-1.68, 5.08] 0.62 0.766 0.19 2. Paired T-test. Since tail is 'one-sided', Pingouin will automatically infer the alternative hypothesis based on the T-value. In the example below, the T-value is negative so the tail is set to 'less'.. >>> 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, tail='one-sided').round(2) T dof tail p-val CI95% cohen-d BF10 power T-test -2.31 4 less 0.04 [-inf, -0.05] 0.25 3.122 0.12 ..which is indeed equivalent to directly testing that x has a smaller mean than y (tail = 'less') >>> ttest(pre, post, paired=True, tail='less').round(2) T dof tail p-val CI95% cohen-d BF10 power T-test -2.31 4 less 0.04 [-inf, -0.05] 0.25 3.122 0.12 3. Now testing the opposite alternative hypothesis (tail = 'greater') >>> ttest(pre, post, paired=True, tail='greater').round(2) T dof tail p-val CI95% cohen-d BF10 power T-test -2.31 4 greater 0.96 [-1.35, inf] 0.25 0.32 0.02 4. 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] >>> stats = ttest(pre, post, paired=True) 5. Independent two-sample T-test (equal sample size). >>> np.random.seed(123) >>> x = np.random.normal(loc=7, size=20) >>> y = np.random.normal(loc=4, size=20) >>> stats = ttest(x, y, correction='auto') 6. Independent two-sample T-test (unequal sample size). >>> np.random.seed(123) >>> x = np.random.normal(loc=7, size=20) >>> y = np.random.normal(loc=6.5, size=15) >>> stats = ttest(x, y, correction='auto') """ from scipy.stats import t, ttest_rel, ttest_ind, ttest_1samp from scipy.stats.stats import (_unequal_var_ttest_denom, _equal_var_ttest_denom) from pingouin import (power_ttest, power_ttest2n, compute_effsize) # Check tails possible_tails = ['two-sided', 'one-sided', 'greater', 'less'] assert tail in possible_tails, 'Invalid tail argument.' 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) 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) 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) # 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) _, se = _equal_var_ttest_denom(vx, nx, vy, ny) # Tail of the test # - Two-sided: default returned by SciPy # - One-sided: 0.5 * two-sided # - Greater / less: (1 - one-sided) or one-sided depending on the means if tail == 'one-sided': # Automatically decide which tail to use based on the T-value tail = 'greater' if tval > 0 else 'less' if tail == 'greater': pval = pval / 2 if tval > 0 else 1 - pval / 2 elif tail == 'less': pval = pval / 2 if tval < 0 else 1 - pval / 2 # Effect size d = compute_effsize(x, y, paired=paired, eftype='cohen') # 95% confidence interval for the effect size # ci = compute_esci(d, nx, ny, paired=paired, eftype='cohen', # confidence=.95) # 95% confidence interval for the difference in means # Compare to the t.test r function conf = 0.975 if tail == 'two-sided' else 0.95 tcrit = t.ppf(conf, dof) ci = np.array([tval - tcrit, tval + tcrit]) * se if tail == 'greater': ci = np.inf elif tail == 'less': ci = -np.inf # Achieved power if ny == 1: # One-sample power = power_ttest(d=d, n=nx, power=None, alpha=0.05, contrast='one-sample', tail=tail) if ny > 1 and paired is True: # Paired two-sample power = power_ttest(d=d, n=nx, power=None, alpha=0.05, contrast='paired', tail=tail) 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', tail=tail) else: # Unequal sample sizes power = power_ttest2n(nx, ny, d=d, power=None, alpha=0.05, tail=tail) # Bayes factor bf = bayesfactor_ttest(tval, nx, ny, paired=paired, tail=tail, r=r) # Create output dictionnary stats = {'dof': np.round(dof, 2), 'T': round(tval, 3), 'p-val': pval, 'tail': tail, 'cohen-d': round(abs(d), 3), 'CI95%': [np.round(ci, 2)], 'power': round(power, 3), 'BF10': bf} # Convert to dataframe col_order = ['T', 'dof', 'tail', 'p-val', 'CI95%', 'cohen-d', 'BF10', 'power'] stats = pd.DataFrame.from_records(stats, columns=col_order, index=['T-test']) return stats
[docs]@pf.register_dataframe_method def rm_anova(data=None, dv=None, within=None, subject=None, correction='auto', detailed=False, export_filename=None): """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 dependant variable (only required if data is in long format). within : 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. export_filename : string Filename (without extension) for the output file. If None, do not export the table. By default, the file will be created in the current python console directory. To change that, specify the filename with full path. Returns ------- aov : 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 'np2' : Partial 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_{total} = SS_{treatment} + (SS_{subjects} + SS_{error}) with .. math:: SS_{total} = \\sum_i^r \\sum_j^n (Y_{ij} - \\overline{Y})^2 .. math:: SS_{treatment} = \\sum_i^r n_i(\\overline{Y_i} - \\overline{Y})^2 .. math:: SS_{subjects} = r\\sum (\\overline{Y}_s - \\overline{Y})^2 .. math:: SS_{error} = SS_{total} - SS_{treatment} - SS_{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_{treatment}}{MS_{error}} = \\frac{\\frac{SS_{treatment}} {r-1}}{\\frac{SS_{error}}{(n - 1)(r - 1)}} and the p-value can be calculated using a F-distribution with :math:v_{treatment} = r - 1 and :math:v_{error} = (n - 1)(r - 1) degrees of freedom. The effect size reported in Pingouin is the partial eta-square, which is equivalent to eta-square for one-way repeated measures ANOVA. .. math:: \\eta_p^2 = \\frac{SS_{treatment}}{SS_{treatment} + SS_{error}} Results have been tested against R and JASP. Note however that if the dataset contains one or more other within subject factors, an automatic collapsing to the mean is applied on the dependant variable (same behavior as the ezANOVA R package). As such, results can differ from those of JASP. Missing values are automatically removed (listwise deletion) using the :py:func:pingouin.remove_rm_na function. This could drastically decrease the power of the ANOVA if many missing values are present. In that case, it might be better to use linear mixed effects models. .. 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. References ---------- ..  Bakeman, R. (2005). Recommended effect size statistics for repeated measures designs. Behavior research methods, 37(3), 379-384. ..  Richardson, J. T. (2011). Eta squared and partial eta squared as measures of effect size in educational research. Educational Research Review, 6(2), 135-147. ..  https://en.wikipedia.org/wiki/Repeated_measures_design Examples -------- 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 np2 eps 0 Within 3 24 5.201 0.006557 0.394 0.694 One-way repeated-measures ANOVA using a long-format dataset >>> df = pg.read_dataset('rm_anova') >>> aov = pg.rm_anova(dv='DesireToKill', within='Disgustingness', ... subject='Subject', data=df, detailed=True) >>> print(aov) Source SS DF MS F p-unc np2 eps 0 Disgustingness 27.485 1 27.485 12.044 0.000793016 0.116 1 1 Error 209.952 92 2.282 - - - - Two-way repeated-measures ANOVA >>> aov = pg.rm_anova(dv='DesireToKill', ... within=['Disgustingness', 'Frighteningness'], ... subject='Subject', data=df) As a :py:class:pandas.DataFrame method >>> df.rm_anova(dv='DesireToKill', within='Disgustingness', ... subject='Subject', detailed=True) Source SS DF MS F p-unc np2 eps 0 Disgustingness 27.485 1 27.485 12.044 0.000793016 0.116 1 1 Error 209.952 92 2.282 - - - - """ if isinstance(within, list): assert len(within) > 0, 'Within is empty.' if len(within) == 1: within = within elif len(within) == 2: return rm_anova2(dv=dv, within=within, data=data, subject=subject, export_filename=export_filename) else: raise ValueError('Repeated measures ANOVA with more than three ' 'factors are not yet supported.') # Check data format if all([v is None for v in [dv, within, subject]]): # Convert from wide to long format assert isinstance(data, pd.DataFrame) data = data._get_numeric_data().dropna() assert data.shape > 2, 'Data must have at least 3 rows.' assert data.shape > 1, 'Data must contain at least two columns.' data['Subj'] = np.arange(data.shape) data = data.melt(id_vars='Subj', var_name='Within', value_name='DV') subject, within, dv = 'Subj', 'Within', 'DV' # Check dataframe _check_dataframe(dv=dv, within=within, data=data, subject=subject, effects='within') # Collapse to the mean data = data.groupby([subject, within]).mean().reset_index() # Remove NaN if data[dv].isnull().any(): data = remove_rm_na(dv=dv, within=within, subject=subject, data=data[[subject, within, dv]]) assert not data[within].isnull().any(), 'Cannot have NaN in within.' assert not data[subject].isnull().any(), 'Cannot have NaN in subject.' # Groupby grp_with = data.groupby(within)[dv] rm = list(data[within].unique()) n_rm = len(rm) n_obs = int(data.groupby(within)[dv].count().max()) grandmean = data[dv].mean() # Calculate sums of squares sstime = ((grp_with.mean() - grandmean)**2 * grp_with.count()).sum() sswithin = grp_with.apply(lambda x: (x - x.mean())**2).sum() grp_subj = data.groupby(subject)[dv] sssubj = n_rm * np.sum((grp_subj.mean() - grandmean)**2) sserror = sswithin - sssubj # Calculate degrees of freedom ddof1 = n_rm - 1 ddof2 = ddof1 * (n_obs - 1) # Calculate F and p-values mserror = sserror / (ddof2 / ddof1) fval = sstime / mserror p_unc = f(ddof1, ddof2).sf(fval) # Calculating partial eta-square # Similar to (fval * ddof1) / (fval * ddof1 + ddof2) np2 = sstime / (sstime + sserror) # Reshape and remove NAN for sphericity estimation and correction data_pivot = data.pivot(index=subject, columns=within, values=dv).dropna() # Compute sphericity using Mauchly test # 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, chi_sq_spher, ddof_spher, \ p_spher = sphericity(data_pivot, alpha=.05) if correction == 'auto': correction = True if not spher else False else: correction = False # Compute epsilon adjustement factor eps = epsilon(data_pivot, correction='gg') # If required, apply Greenhouse-Geisser correction for sphericity if correction: corr_ddof1, corr_ddof2 = [np.maximum(d * eps, 1.) 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, 'np2': np2, 'eps': eps, }, index=) 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', 'np2', 'eps', 'sphericity', 'W-spher', 'p-spher'] else: aov = pd.DataFrame({'Source': [within, 'Error'], 'SS': np.round([sstime, sserror], 3), 'DF': [ddof1, ddof2], 'MS': np.round([sstime / ddof1, sserror / ddof2], 3), 'F': [fval, np.nan], 'p-unc': [p_unc, np.nan], 'np2': [np2, 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', 'np2', 'eps', 'sphericity', 'W-spher', 'p-spher'] # Round aov[['F', 'eps', 'np2']] = aov[['F', 'eps', 'np2']].round(3) # Replace NaN aov = aov.fillna('-') aov = aov.reindex(columns=col_order) aov.dropna(how='all', axis=1, inplace=True) # Export to .csv if export_filename is not None: _export_table(aov, export_filename) return aov
def rm_anova2(data=None, dv=None, within=None, subject=None, export_filename=None): """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. Parameters ---------- data : :py:class:pandas.DataFrame DataFrame dv : string Name of column containing the dependant variable. within : list Names of column containing the two within factor (e.g. ['Time', 'Treatment']) subject : string Name of column containing the subject identifier. export_filename : string Filename (without extension) for the output file. If None, do not export the table. By default, the file will be created in the current python console directory. To change that, specify the filename with full path. Returns ------- aov : DataFrame ANOVA summary :: 'Source' : Name of the within-group factors 'ddof1' : Degrees of freedom (numerator) 'ddof2' : Degrees of freedom (denominator) 'F' : F-value 'p-unc' : Uncorrected p-value 'np2' : Partial eta-square effect size 'eps' : Greenhouse-Geisser epsilon factor (= index of sphericity) 'p-GG-corr' : Greenhouse-Geisser corrected p-value """ a, b = within # Validate the dataframe _check_dataframe(dv=dv, within=within, data=data, subject=subject, effects='within') # Remove NaN if data[[subject, a, b, dv]].isnull().any().any(): data = remove_rm_na(dv=dv, subject=subject, within=[a, b], data=data[[subject, a, b, dv]]) # Collapse to the mean (that this is also done in remove_rm_na) data = data.groupby([subject, a, b]).mean().reset_index() assert not data[a].isnull().any(), 'Cannot have NaN in %s' % a assert not data[b].isnull().any(), 'Cannot have NaN in %s' % b assert not data[subject].isnull().any(), 'Cannot have NaN in %s' % subject # Group sizes and grandmean n_a = data[a].nunique() n_b = data[b].nunique() n_s = data[subject].nunique() mu = data[dv].mean() # Groupby means grp_s = data.groupby(subject)[dv].mean() grp_a = data.groupby([a])[dv].mean() grp_b = data.groupby([b])[dv].mean() grp_ab = data.groupby([a, b])[dv].mean() grp_as = data.groupby([a, subject])[dv].mean() grp_bs = data.groupby([b, subject])[dv].mean() # 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) # Partial eta-square eta_a = (f_a * df_a) / (f_a * df_a + df_as) eta_b = (f_b * df_b) / (f_b * df_b + df_bs) eta_ab = (f_ab * df_ab) / (f_ab * df_ab + df_abs) # Epsilon piv_a = data.pivot_table(index=subject, columns=a, values=dv) piv_b = data.pivot_table(index=subject, columns=b, values=dv) piv_ab = data.pivot_table(index=subject, columns=[a, b], values=dv) 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(piv_ab, correction='gg') # Greenhouse-Geisser correction df_a_c, df_as_c = [np.maximum(d * eps_a, 1.) for d in (df_a, df_as)] df_b_c, df_bs_c = [np.maximum(d * eps_b, 1.) for d in (df_b, df_bs)] df_ab_c, df_abs_c = [np.maximum(d * eps_ab, 1.) 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], 'np2': [eta_a, eta_b, eta_ab], 'eps': [eps_a, eps_b, eps_ab], }) col_order = ['Source', 'SS', 'ddof1', 'ddof2', 'MS', 'F', 'p-unc', 'p-GG-corr', 'np2', 'eps'] # Round aov[['SS', 'MS', 'F', 'eps', 'np2']] = aov[['SS', 'MS', 'F', 'eps', 'np2']].round(3) aov = aov.reindex(columns=col_order) # Export to .csv if export_filename is not None: _export_table(aov, export_filename) return aov
[docs]@pf.register_dataframe_method def anova(data=None, dv=None, between=None, ss_type=2, detailed=False, export_filename=None): """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). export_filename : string Filename (without extension) for the output file. If None, do not export the table. By default, the file will be created in the current python console directory. To change that, specify the filename with full path. Returns ------- aov : 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_{total} = SS_{treatment} + SS_{error} .. math:: SS_{total} = \\sum_i \\sum_j (Y_{ij} - \\overline{Y})^2 .. math:: SS_{treatment} = \\sum_i n_i (\\overline{Y_i} - \\overline{Y})^2 .. math:: SS_{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_{treatment}}{MS_{error}} = \\frac{SS_{treatment} / (r - 1)}{SS_{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 effect size reported in Pingouin is the partial eta-square. However, one should keep in mind that for one-way ANOVA partial eta-square is the same as eta-square and generalized eta-square. For more details, see Bakeman 2005; Richardson 2011. .. math:: \\eta_p^2 = \\frac{SS_{treatment}}{SS_{treatment} + SS_{error}} Note that missing values are automatically removed. Results have been tested against R, Matlab and JASP. .. warning :: Versions of Pingouin below 0.2.5 gave wrong results for **unbalanced N-way ANOVA**. This issue has been resolved in Pingouin>=0.2.5. In such cases, the ANOVA is calculated via an internal call to the statsmodels package. References ---------- ..  Liu, Hangcheng. "Comparing Welch's ANOVA, a Kruskal-Wallis test and traditional ANOVA in case of Heterogeneity of Variance." (2015). ..  Bakeman, Roger. "Recommended effect size statistics for repeated measures designs." Behavior research methods 37.3 (2005): 379-384. ..  Richardson, John TE. "Eta squared and partial eta squared as measures of effect size in educational research." Educational Research Review 6.2 (2011): 135-147. 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 Source SS DF MS F p-unc np2 0 Hair color 1360.726 3 453.575 6.791 0.00411423 0.576 1 Within 1001.800 15 66.787 - - - Note that this function can also directly be used as a Pandas method >>> df.anova(dv='Pain threshold', between='Hair color', detailed=True) Source SS DF MS F p-unc np2 0 Hair color 1360.726 3 453.575 6.791 0.00411423 0.576 1 Within 1001.800 15 66.787 - - - 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"]).round(3) Source SS DF MS F p-unc np2 0 Diet 390.625 1.0 390.625 7.423 0.034 0.553 1 Exercise 180.625 1.0 180.625 3.432 0.113 0.364 2 Diet * Exercise 15.625 1.0 15.625 0.297 0.605 0.047 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) Source SS DF MS F p-unc np2 0 Sex 2.075 1.0 2.075 2.462 0.123191 0.049 1 Risk 11.332 1.0 11.332 13.449 0.000613 0.219 2 Drug 0.816 2.0 0.408 0.484 0.619249 0.020 3 Sex * Risk 0.117 1.0 0.117 0.139 0.710541 0.003 4 Sex * Drug 2.564 2.0 1.282 1.522 0.228711 0.060 5 Risk * Drug 2.438 2.0 1.219 1.446 0.245485 0.057 6 Sex * Risk * Drug 1.844 2.0 0.922 1.094 0.343041 0.044 7 Residual 40.445 48.0 0.843 NaN NaN NaN """ if isinstance(between, list): if len(between) == 0: raise ValueError('between is empty.') elif len(between) == 1: between = between 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, export_filename=export_filename) else: # 3 or more factors with (un)-balanced design = statsmodels return anovan(dv=dv, between=between, data=data, ss_type=ss_type, export_filename=export_filename) # Check 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) groups = list(data[between].unique()) n_groups = len(groups) N = data[dv].size # Calculate sums of squares grp = data.groupby(between)[dv] # Between effect ssbetween = ((grp.mean() - data[dv].mean())**2 * grp.count()).sum() # Within effect (= error between) # = (grp.var(ddof=0) * grp.count()).sum() sserror = grp.apply(lambda x: (x - x.mean())**2).sum() # 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 partial eta-square # Similar to (fval * ddof1) / (fval * ddof1 + ddof2) np2 = ssbetween / (ssbetween + sserror) # Create output dataframe if not detailed: aov = pd.DataFrame({'Source': between, 'ddof1': ddof1, 'ddof2': ddof2, 'F': fval, 'p-unc': p_unc, 'np2': np2 }, index=) col_order = ['Source', 'ddof1', 'ddof2', 'F', 'p-unc', 'np2'] else: aov = pd.DataFrame({'Source': [between, 'Within'], 'SS': np.round([ssbetween, sserror], 3), 'DF': [ddof1, ddof2], 'MS': np.round([msbetween, mserror], 3), 'F': [fval, np.nan], 'p-unc': [p_unc, np.nan], 'np2': [np2, np.nan] }) col_order = ['Source', 'SS', 'DF', 'MS', 'F', 'p-unc', 'np2'] # Round aov[['F', 'np2']] = aov[['F', 'np2']].round(3) # Replace NaN aov = aov.fillna('-') aov = aov.reindex(columns=col_order) aov.dropna(how='all', axis=1, inplace=True) # Export to .csv if export_filename is not None: _export_table(aov, export_filename) return aov
def anova2(data=None, dv=None, between=None, ss_type=2, export_filename=None): """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 _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 >= 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)[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())**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, export_filename=export_filename) # 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) # 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) # Create output dataframe aov = pd.DataFrame({'Source': [fac1, fac2, fac1 + ' * ' + fac2, 'Residual'], 'SS': np.round([ss_fac1, ss_fac2, ss_inter, ss_resid], 3), 'DF': [df_fac1, df_fac2, df_inter, df_resid], 'MS': np.round([ms_fac1, ms_fac2, ms_inter, ms_resid], 3), 'F': [fval_fac1, fval_fac2, fval_inter, np.nan], 'p-unc': [pval_fac1, pval_fac2, pval_inter, np.nan], 'np2': [np2_fac1, np2_fac2, np2_inter, np.nan] }) col_order = ['Source', 'SS', 'DF', 'MS', 'F', 'p-unc', 'np2'] aov = aov.reindex(columns=col_order) aov.dropna(how='all', axis=1, inplace=True) # Export to .csv if export_filename is not None: _export_table(aov, export_filename) return aov def anovan(data=None, dv=None, between=None, ss_type=2, export_filename=None): """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 _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 >= 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 formula = dv + ' ~ ' for fac in between: formula += 'C(' + fac + ', Sum) * ' 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'] 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(%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', 'np2'] aov = aov.reindex(columns=col_order) aov[['SS', 'MS', 'F', 'np2']] = aov[['SS', 'MS', 'F', 'np2']].round(3) aov.dropna(how='all', axis=1, inplace=True) # Add formula to dataframe aov.formula_ = formula # Export to .csv if export_filename is not None: _export_table(aov, export_filename) return aov
[docs]@pf.register_dataframe_method def welch_anova(data=None, dv=None, between=None, export_filename=None): """One-way Welch ANOVA. Parameters ---------- dv : string Name of column containing the dependant variable. between : string Name of column containing the between factor. 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. export_filename : string Filename (without extension) for the output file. If None, do not export the table. By default, the file will be created in the current python console directory. To change that, specify the filename with full path. Returns ------- aov : 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 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}_{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 treatment sums of squares is defined as: .. math:: SS_{treatment} = \\sum_{i=1}^r w_i (\\overline{Y}_i - \\overline{Y}_{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_{welch} = \\frac{SS_{treatment} / (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 ---------- ..  Liu, Hangcheng. "Comparing Welch's ANOVA, a Kruskal-Wallis test and traditional ANOVA in case of Heterogeneity of Variance." (2015). ..  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, export_filename='pain_anova.csv') >>> aov Source ddof1 ddof2 F p-unc 0 Hair color 3 8.33 5.89 0.018813 """ # Check 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)[dv] weights = grp.count() / grp.var() adj_grandmean = (weights * grp.mean()).sum() / weights.sum() # Treatment sum of squares ss_tr = np.sum(weights * np.square(grp.mean() - adj_grandmean)) ms_tr = ss_tr / ddof1 # Calculate lambda, F-value and p-value lamb = (3 * np.sum((1 / (grp.count() - 1)) * (1 - (weights / weights.sum()))**2)) / (r**2 - 1) fval = ms_tr / (1 + (2 * lamb * (r - 2)) / 3) pval = f.sf(fval, ddof1, 1 / lamb) # Create output dataframe aov = pd.DataFrame({'Source': between, 'ddof1': ddof1, 'ddof2': 1 / lamb, 'F': fval, 'p-unc': pval, }, index=) col_order = ['Source', 'ddof1', 'ddof2', 'F', 'p-unc'] aov = aov.reindex(columns=col_order) aov[['F', 'ddof2']] = aov[['F', 'ddof2']].round(3) # Export to .csv if export_filename is not None: _export_table(aov, export_filename) return aov
[docs]@pf.register_dataframe_method def mixed_anova(data=None, dv=None, within=None, subject=None, between=None, correction='auto', export_filename=None): """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 dependant variable. within : string Name of column containing the within factor. subject : string Name of column containing the 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. export_filename : string Filename (without extension) for the output file. If None, do not export the table. By default, the file will be created in the current python console directory. To change that, specify the filename with full path. Returns ------- aov : 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-square 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 : One-way and N-way ANOVA rm_anova : One-way and two-way repeated measures ANOVA Notes ----- Results have been tested against R and JASP. Missing values are automatically removed (listwise deletion) using the :py:func:pingouin.remove_rm_na function. This could drastically decrease the power of the ANOVA if many missing values are present. In that case, it might be better to use linear mixed effects models. If the between-subject groups are unbalanced (= unequal sample sizes), a type II ANOVA will be computed. Examples -------- 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 Source SS DF1 DF2 MS F p-unc np2 eps 0 Group 5.460 1 58 5.460 5.052 0.028420 0.080 - 1 Time 7.628 2 116 3.814 4.027 0.020373 0.065 0.999 2 Interaction 5.168 2 116 2.584 2.728 0.069530 0.045 - """ # Check data _check_dataframe(dv=dv, within=within, between=between, data=data, subject=subject, effects='interaction') # Collapse to the mean data = data.groupby([subject, within, between]).mean().reset_index() # Remove NaN if data[dv].isnull().any(): data = remove_rm_na(dv=dv, within=within, subject=subject, data=data[[subject, within, between, dv]]) # SUMS OF SQUARES grandmean = data[dv].mean() # Extract main effects of time and between mtime = rm_anova(dv=dv, within=within, subject=subject, data=data, correction=correction, detailed=True) mbetw = anova(dv=dv, between=between, data=data, detailed=True) # Extract SS total, residuals and interactions grp = data.groupby([between, within])[dv] sstotal = grp.apply(lambda x: (x - grandmean)**2).sum() # sst = residuals within + residuals between sst = grp.apply(lambda x: (x - x.mean())**2).sum() # Interaction ssinter = sstotal - (sst + mtime.at[0, 'SS'] + mbetw.at[0, 'SS']) sswg = mtime.at[1, 'SS'] - ssinter sseb = sstotal - (mtime.at[0, 'SS'] + mbetw.at[0, 'SS'] + sswg + ssinter) # DEGREES OF FREEDOM n_obs = data.groupby(within)[dv].count().max() dftime = mtime.at[0, 'DF'] dfbetween = mbetw.at[0, 'DF'] dfeb = n_obs - data.groupby(between)[dv].count().count() dfwg = dftime * dfeb dfinter = mtime.at[0, 'DF'] * mbetw.at[0, 'DF'] # MEAN SQUARES mseb = sseb / dfeb mswg = sswg / dfwg msinter = ssinter / dfinter # F VALUES fbetween = mbetw.at[0, 'MS'] / mseb ftime = mtime.at[0, 'MS'] / mswg finter = msinter / mswg # P-values pbetween = f(dfbetween, dfeb).sf(fbetween) ptime = f(dftime, dfwg).sf(ftime) pinter = f(dfinter, dfwg).sf(finter) # Effects sizes npsq_between = fbetween * dfbetween / (fbetween * dfbetween + dfeb) npsq_time = ftime * dftime / (ftime * dftime + dfwg) npsq_inter = ssinter / (ssinter + sswg) # Stats table aov = pd.concat([mbetw.drop(1), mtime.drop(1)], sort=False, ignore_index=True) # Update values aov.rename(columns={'DF': 'DF1'}, inplace=True) aov.at[0, 'F'], aov.at[1, 'F'] = fbetween, ftime aov.at[0, 'p-unc'], aov.at[1, 'p-unc'] = pbetween, ptime aov.at[0, 'np2'], aov.at[1, 'np2'] = npsq_between, npsq_time aov = aov.append({'Source': 'Interaction', 'SS': ssinter, 'DF1': dfinter, 'MS': msinter, 'F': finter, 'p-unc': pinter, 'np2': npsq_inter, }, ignore_index=True) aov['SS'] = aov['SS'].round(3) aov['MS'] = aov['MS'].round(3) aov['DF2'] = [dfeb, dfwg, dfwg] aov['eps'] = [np.nan, mtime.at[0, 'eps'], np.nan] col_order = ['Source', 'SS', 'DF1', 'DF2', 'MS', 'F', 'p-unc', 'p-GG-corr', 'np2', 'eps', 'sphericity', 'W-spher', 'p-spher'] # Replace NaN aov = aov.fillna('-') aov = aov.reindex(columns=col_order) aov.dropna(how='all', axis=1, inplace=True) # Round aov[['F', 'eps', 'np2']] = aov[['F', 'eps', 'np2']].round(3) # Export to .csv if export_filename is not None: _export_table(aov, export_filename) return aov
[docs]@pf.register_dataframe_method def ancova(data=None, dv=None, between=None, covar=None, export_filename=None): """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 containing the dependant variable. between : string Name of column containing the between factor. covar : string or list Name(s) of column(s) containing the covariate. export_filename : string Filename (without extension) for the output file. If None, do not export the table. By default, the file will be created in the current python console directory. To change that, specify the filename with full path. Returns ------- aov : 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 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). Note that in the case of one covariate, Pingouin will use a built-in function. However, if there are more than one covariate, Pingouin will use the statsmodels package to compute the ANCOVA. 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 0 Method 571.030045 3 3.336482 0.031940 1 Income 1678.352687 1 29.419438 0.000006 2 Residual 1768.522365 31 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') Source SS DF F p-unc 0 Method 552.284 3 3.233 0.036113 1 Income 1573.952 1 27.637 0.000011 2 BMI 60.014 1 1.054 0.312842 3 Residual 1708.509 30 NaN NaN """ # Safety checks assert isinstance(data, pd.DataFrame) 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.' # Drop missing values data = data[_flatten_list([dv, between, covar])].dropna() # Check the number of covariates if isinstance(covar, list): if len(covar) > 1: return ancovan(dv=dv, covar=covar, between=between, data=data, export_filename=export_filename) else: covar = covar # Assert that covariate is numeric assert data[covar].dtype.kind in 'fi', 'Covariate must be numeric.' def linreg(x, y): return np.corrcoef(x, y)[0, 1] * np.std(y, ddof=1) / np.std(x, ddof=1) # Compute slopes groups = data[between].unique() slopes = np.zeros(shape=groups.shape) ss = np.zeros(shape=groups.shape) for i, b in enumerate(groups): dt_covar = data[data[between] == b][covar].values ss[i] = ((dt_covar - dt_covar.mean())**2).sum() slopes[i] = linreg(dt_covar, data[data[between] == b][dv].values) ss_slopes = ss * slopes bw = np.nansum(ss_slopes) / np.sum(ss) bt = linreg(data[covar], data[dv]) # Run the ANOVA aov_dv = anova(data=data, dv=dv, between=between, detailed=True) aov_covar = anova(data=data, dv=covar, between=between, detailed=True) # Create full ANCOVA ss_t_dv = aov_dv.at[0, 'SS'] + aov_dv.at[1, 'SS'] ss_t_covar = aov_covar.at[0, 'SS'] + aov_covar.at[1, 'SS'] # Sums of squares ss_t = ss_t_dv - bt**2 * ss_t_covar ss_w = aov_dv.at[1, 'SS'] - bw**2 * aov_covar.at[1, 'SS'] ss_b = ss_t - ss_w ss_c = np.nansum(ss_slopes) * bw # DOF df_c = 1 df_b = aov_dv.at[0, 'DF'] df_w = aov_dv.at[1, 'DF'] - 1 # Mean squares ms_c = ss_c / df_c ms_b = ss_b / df_b ms_w = ss_w / df_w # F-values f_c = ms_c / ms_w f_b = ms_b / ms_w # P-values p_c = f(df_c, df_w).sf(f_c) p_b = f(df_b, df_w).sf(f_b) # Create dataframe aov = pd.DataFrame({'Source': [between, covar, 'Residual'], 'SS': [ss_b, ss_c, ss_w], 'DF': [df_b, df_c, df_w], 'F': [f_b, f_c, np.nan], 'p-unc': [p_b, p_c, np.nan], }) # Export to .csv if export_filename is not None: _export_table(aov, export_filename) # Add bw as an attribute (for rm_corr function) aov.bw_ = bw return aov
def ancovan(data=None, dv=None, covar=None, between=None, export_filename=None): """ANCOVA with n covariates. This is an internal function. The main call to this function should be done by the :py:func:pingouin.ancova function. Parameters ---------- data : :py:class:pandas.DataFrame DataFrame dv : string Name of column containing the dependant variable. covar : string Name(s) of columns containing the covariates. between : string Name of column containing the between factor. export_filename : string Filename (without extension) for the output file. If None, do not export the table. By default, the file will be created in the current python console directory. To change that, specify the filename with full path. Returns ------- aov : 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 """ # 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 # Check that covariates are numeric ('float', 'int') assert all([data[covar[i]].dtype.kind in 'fi' for i in range(len(covar))]) # Fit ANCOVA model formula = dv + ' ~ C(' + between + ')' for c in covar: formula += ' + ' + c model = ols(formula, data=data).fit() 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 aov['DF'] = aov['DF'].astype(int) aov[['SS', 'F']] = aov[['SS', 'F']].round(3) # Export to .csv if export_filename is not None: _export_table(aov, export_filename) return aov