Source code for pingouin.reliability

import numpy as np
import pandas as pd
from scipy.stats import f

__all__ = ["cronbach_alpha", "intraclass_corr"]


[docs]def cronbach_alpha(data=None, items=None, scores=None, subject=None, nan_policy='pairwise', ci=.95): """Cronbach's alpha reliability measure. Parameters ---------- data : pandas dataframe Wide or long-format dataframe. items : str Column in ``data`` with the items names (long-format only). scores : str Column in ``data`` with the scores (long-format only). subject : str Column in ``data`` with the subject identifier (long-format only). nan_policy : bool If `'listwise'`, remove the entire rows that contain missing values (= listwise deletion). If `'pairwise'` (default), only pairwise missing values are removed when computing the covariance matrix. For more details, please refer to the :py:meth:`pandas.DataFrame.cov` method. ci : float Confidence interval (.95 = 95%) Returns ------- alpha : float Cronbach's alpha Notes ----- This function works with both wide and long format dataframe. If you pass a long-format dataframe, you must also pass the ``items``, ``scores`` and ``subj`` columns (in which case the data will be converted into wide format using the :py:meth:`pandas.DataFrame.pivot` method). Internal consistency is usually measured with Cronbach's alpha, a statistic calculated from the pairwise correlations between items. Internal consistency ranges between negative infinity and one. Coefficient alpha will be negative whenever there is greater within-subject variability than between-subject variability. Cronbach's :math:`\\alpha` is defined as .. math:: \\alpha ={k \\over k-1}\\left(1-{\\sum_{{i=1}}^{k}\\sigma_{{y_{i}}}^{2} \\over\\sigma_{x}^{2}}\\right) where :math:`k` refers to the number of items, :math:`\\sigma_{x}^{2}` is the variance of the observed total scores, and :math:`\\sigma_{{y_{i}}}^{2}` the variance of component :math:`i` for the current sample of subjects. Another formula for Cronbach's :math:`\\alpha` is .. math:: \\alpha = \\frac{k \\times \\bar c}{\\bar v + (k - 1) \\times \\bar c} where :math:`\\bar c` refers to the average of all covariances between items and :math:`\\bar v` to the average variance of each item. 95% confidence intervals are calculated using Feldt's method: .. math:: c_L = 1 - (1 - \\alpha) \\cdot F_{(0.025, n-1, (n-1)(k-1))} c_U = 1 - (1 - \\alpha) \\cdot F_{(0.975, n-1, (n-1)(k-1))} where :math:`n` is the number of subjects and :math:`k` the number of items. Results have been tested against the R package psych. References ---------- .. [1] https://en.wikipedia.org/wiki/Cronbach%27s_alpha .. [2] http://www.real-statistics.com/reliability/cronbachs-alpha/ .. [3] https://cran.r-project.org/web/packages/psych/psych.pdf .. [4] Feldt, Leonard S., Woodruff, David J., & Salih, Fathi A. (1987). Statistical inference for coefficient alpha. Applied Psychological Measurement, 11(1):93-103. Examples -------- Binary wide-format dataframe (with missing values) >>> import pingouin as pg >>> data = pg.read_dataset('cronbach_wide_missing') >>> # In R: psych:alpha(data, use="pairwise") >>> pg.cronbach_alpha(data=data) (0.732661, array([0.435, 0.909])) After listwise deletion of missing values (remove the entire rows) >>> # In R: psych:alpha(data, use="complete.obs") >>> pg.cronbach_alpha(data=data, nan_policy='listwise') (0.801695, array([0.581, 0.933])) After imputing the missing values with the median of each column >>> pg.cronbach_alpha(data=data.fillna(data.median())) (0.738019, array([0.447, 0.911])) Likert-type long-format dataframe >>> data = pg.read_dataset('cronbach_alpha') >>> pg.cronbach_alpha(data=data, items='Items', scores='Scores', ... subject='Subj') (0.591719, array([0.195, 0.84 ])) """ # Safety check assert isinstance(data, pd.DataFrame), 'data must be a dataframe.' assert nan_policy in ['pairwise', 'listwise'] if all([v is not None for v in [items, scores, subject]]): # Data in long-format: we first convert to a wide format data = data.pivot(index=subject, values=scores, columns=items) # From now we assume that data is in wide format n, k = data.shape assert k >= 2, 'At least two items are required.' assert n >= 2, 'At least two raters/subjects are required.' err = 'All columns must be numeric.' assert all([data[c].dtype.kind in 'bfi' for c in data.columns]), err if data.isna().any().any() and nan_policy == 'listwise': # In R = psych:alpha(data, use="complete.obs") data = data.dropna(axis=0, how='any') # Compute covariance matrix and Cronbach's alpha C = data.cov() cronbach = (k / (k - 1)) * (1 - np.trace(C) / C.sum().sum()) # which is equivalent to # v = np.diag(C).mean() # c = C.values[np.tril_indices_from(C, k=-1)].mean() # cronbach = (k * c) / (v + (k - 1) * c) # Confidence intervals alpha = 1 - ci df1 = n - 1 df2 = df1 * (k - 1) lower = 1 - (1 - cronbach) * f.isf(alpha / 2, df1, df2) upper = 1 - (1 - cronbach) * f.isf(1 - alpha / 2, df1, df2) return round(cronbach, 6), np.round([lower, upper], 3)
[docs]def intraclass_corr(data=None, targets=None, raters=None, ratings=None, nan_policy='raise'): """Intraclass correlation. Parameters ---------- data : :py:class:`pandas.DataFrame` Long-format dataframe. Data must be fully balanced. targets : string Name of column in ``data`` containing the targets. raters : string Name of column in ``data`` containing the raters. ratings : string Name of column in ``data`` containing the ratings. nan_policy : str Defines how to handle when input contains missing values (nan). 'raise' (default) throws an error, 'omit' performs the calculations after deleting target(s) with one or more missing values (= listwise deletion). .. versionadded:: 0.3.0 Returns ------- stats : :py:class:`pandas.DataFrame` Output dataframe: * ``'Type'``: ICC type * ``'Description'``: description of the ICC * ``'ICC'``: intraclass correlation * ``'F'``: F statistic * ``'df1'``: numerator degree of freedom * ``'df2'``: denominator degree of freedom * ``'pval'``: p-value * ``'CI95%'``: 95% confidence intervals around the ICC Notes ----- The intraclass correlation (ICC) assesses the reliability of ratings by comparing the variability of different ratings of the same subject to the total variation across all ratings and all subjects. Shrout and Fleiss (1979) describe six cases of reliability of ratings done by :math:`k` raters on :math:`n` targets. Pingouin returns all six cases with corresponding F and p-values, as well as 95% confidence intervals. From the documentation of the ICC function in the R package psych: - **ICC1**: Each target is rated by a different rater and the raters are selected at random. This is a one-way ANOVA fixed effects model. - **ICC2**: A random sample of :math:`k` raters rate each target. The measure is one of absolute agreement in the ratings. ICC1 is sensitive to differences in means between raters and is a measure of absolute agreement. - **ICC3**: A fixed set of :math:`k` raters rate each target. There is no generalization to a larger population of raters. ICC2 and ICC3 remove mean differences between raters, but are sensitive to interactions. The difference between ICC2 and ICC3 is whether raters are seen as fixed or random effects. Then, for each of these cases, the reliability can either be estimated for a single rating or for the average of :math:`k` ratings. The 1 rating case is equivalent to the average intercorrelation, while the :math:`k` rating case is equivalent to the Spearman Brown adjusted reliability. **ICC1k**, **ICC2k**, **ICC3K** reflect the means of :math:`k` raters. This function has been tested against the ICC function of the R psych package. Note however that contrarily to the R implementation, the current implementation does not use linear mixed effect but regular ANOVA, which means that it only works with complete-case data (no missing values). References ---------- .. [1] Shrout, P. E., & Fleiss, J. L. (1979). Intraclass correlations: uses in assessing rater reliability. Psychological bulletin, 86(2), 420. .. [2] https://cran.r-project.org/web/packages/psych/psych.pdf .. [3] http://www.real-statistics.com/reliability/intraclass-correlation/ Examples -------- ICCs of wine quality assessed by 4 judges. >>> import pingouin as pg >>> data = pg.read_dataset('icc') >>> icc = pg.intraclass_corr(data=data, targets='Wine', raters='Judge', ... ratings='Scores') >>> icc # doctest: +SKIP Type Description ICC ... df2 pval CI95% 0 ICC1 Single raters absolute 0.728 ... 24 0.000002 [0.43, 0.93] 1 ICC2 Single random raters 0.728 ... 21 0.000005 [0.43, 0.93] 2 ICC3 Single fixed raters 0.730 ... 21 0.000005 [0.43, 0.93] 3 ICC1k Average raters absolute 0.914 ... 24 0.000002 [0.75, 0.98] 4 ICC2k Average random raters 0.914 ... 21 0.000005 [0.75, 0.98] 5 ICC3k Average fixed raters 0.915 ... 21 0.000005 [0.75, 0.98] """ from pingouin import anova # Safety check assert isinstance(data, pd.DataFrame), 'data must be a dataframe.' assert all([v is not None for v in [targets, raters, ratings]]) assert all([v in data.columns for v in [targets, raters, ratings]]) assert nan_policy in ['omit', 'raise'] # Convert data to wide-format data = data.pivot_table(index=targets, columns=raters, values=ratings) # Listwise deletion of missing values nan_present = data.isna().any().any() if nan_present: if nan_policy == 'omit': data = data.dropna(axis=0, how='any') else: raise ValueError("Either missing values are present in data or " "data are unbalanced. Please remove them " "manually or use nan_policy='omit'.") # Back to long-format # data_wide = data.copy() # Optional, for PCA data = data.reset_index().melt(id_vars=targets, value_name=ratings) # Check that ratings is a numeric variable assert data[ratings].dtype.kind in 'bfi', 'Ratings must be numeric.' # Check that data are fully balanced # This behavior is ensured by the long-to-wide-to-long transformation # Unbalanced data will result in rows with missing values. # assert data.groupby(raters)[ratings].count().nunique() == 1 # Extract sizes k = data[raters].nunique() n = data[targets].nunique() # Two-way ANOVA with np.errstate(invalid='ignore'): aov = anova(dv=ratings, between=[targets, raters], data=data, ss_type=2) # Extract mean squares msb = aov.at[0, 'MS'] msw = (aov.at[1, 'SS'] + aov.at[2, 'SS']) / (aov.at[1, 'DF'] + aov.at[2, 'DF']) msj = aov.at[1, 'MS'] mse = aov.at[2, 'MS'] # Calculate ICCs icc1 = (msb - msw) / (msb + (k - 1) * msw) icc2 = (msb - mse) / (msb + (k - 1) * mse + k * (msj - mse) / n) icc3 = (msb - mse) / (msb + (k - 1) * mse) icc1k = (msb - msw) / msb icc2k = (msb - mse) / (msb + (msj - mse) / n) icc3k = (msb - mse) / msb # Calculate F, df, and p-values f1k = msb / msw df1 = n - 1 df1kd = n * (k - 1) p1k = f.sf(f1k, df1, df1kd) f2k = f3k = msb / mse df2kd = (n - 1) * (k - 1) p2k = f.sf(f2k, df1, df2kd) # Create output dataframe stats = { 'Type': ['ICC1', 'ICC2', 'ICC3', 'ICC1k', 'ICC2k', 'ICC3k'], 'Description': ['Single raters absolute', 'Single random raters', 'Single fixed raters', 'Average raters absolute', 'Average random raters', 'Average fixed raters'], 'ICC': np.round([icc1, icc2, icc3, icc1k, icc2k, icc3k], 3), 'F': np.round([f1k, f2k, f2k, f1k, f2k, f2k], 3), 'df1': n - 1, 'df2': [df1kd, df2kd, df2kd, df1kd, df2kd, df2kd], 'pval': [p1k, p2k, p2k, p1k, p2k, p2k] } stats = pd.DataFrame(stats) # Calculate confidence intervals alpha = 0.05 # Case 1 and 3 f1l = f1k / f.ppf(1 - alpha / 2, df1, df1kd) f1u = f1k * f.ppf(1 - alpha / 2, df1kd, df1) l1 = (f1l - 1) / (f1l + (k - 1)) u1 = (f1u - 1) / (f1u + (k - 1)) f3l = f3k / f.ppf(1 - alpha / 2, df1, df2kd) f3u = f3k * f.ppf(1 - alpha / 2, df2kd, df1) l3 = (f3l - 1) / (f3l + (k - 1)) u3 = (f3u - 1) / (f3u + (k - 1)) # Case 2 fj = msj / mse vn = df2kd * ((k * icc2 * fj + n * (1 + (k - 1) * icc2) - k * icc2))**2 vd = df1 * k**2 * icc2**2 * fj**2 + \ (n * (1 + (k - 1) * icc2) - k * icc2)**2 v = vn / vd f2u = f.ppf(1 - alpha / 2, n - 1, v) f2l = f.ppf(1 - alpha / 2, v, n - 1) l2 = n * (msb - f2u * mse) / (f2u * (k * msj + (k * n - k - n) * mse) + n * msb) u2 = n * (f2l * msb - mse) / (k * msj + (k * n - k - n) * mse + n * f2l * msb) # Round the confidence intervals def list_round(x, decimals=2): for i, xi in enumerate(x): x[i] = np.round(xi, decimals).tolist() return x stats['CI95%'] = list_round([[l1, u1], [l2, u2], [l3, u3], [1 - 1 / f1l, 1 - 1 / f1u], [l2 * k / (1 + l2 * (k - 1)), u2 * k / (1 + u2 * (k - 1))], [1 - 1 / f3l, 1 - 1 / f3u]]) return stats