Source code for statsmodels.imputation.mice

"""
Overview
--------

This module implements the Multiple Imputation through Chained
Equations (MICE) approach to handling missing data in statistical data
analyses. The approach has the following steps:

0. Impute each missing value with the mean of the observed values of
the same variable.

1. For each variable in the data set with missing values (termed the
'focus variable'), do the following:

1a. Fit an 'imputation model', which is a regression model for the
focus variable, regressed on the observed and (current) imputed values
of some or all of the other variables.

1b. Impute the missing values for the focus variable.  Currently this
imputation must use the 'predictive mean matching' (pmm) procedure.

2. Once all variables have been imputed, fit the 'analysis model' to
the data set.

3. Repeat steps 1-2 multiple times and combine the results using a
'combining rule' to produce point estimates of all parameters in the
analysis model and standard errors for them.

The imputations for each variable are based on an imputation model
that is specified via a model class and a formula for the regression
relationship.  The default model is OLS, with a formula specifying
main effects for all other variables.

The MICE procedure can be used in one of two ways:

* If the goal is only to produce imputed data sets, the MICEData class
can be used to wrap a data frame, providing facilities for doing the
imputation.  Summary plots are available for assessing the performance
of the imputation.

* If the imputed data sets are to be used to fit an additional
'analysis model', a MICE instance can be used.  After specifying the
MICE instance and running it, the results are combined using the
`combine` method.  Results and various summary plots are then
available.

Terminology
-----------

The primary goal of the analysis is usually to fit and perform
inference using an 'analysis model'. If an analysis model is not
specified, then imputed datasets are produced for later use.

The MICE procedure involves a family of imputation models.  There is
one imputation model for each variable with missing values.  An
imputation model may be conditioned on all or a subset of the
remaining variables, using main effects, transformations,
interactions, etc. as desired.

A 'perturbation method' is a method for setting the parameter estimate
in an imputation model.  The 'gaussian' perturbation method first fits
the model (usually using maximum likelihood, but it could use any
statsmodels fit procedure), then sets the parameter vector equal to a
draw from the Gaussian approximation to the sampling distribution for
the fit.  The 'bootstrap' perturbation method sets the parameter
vector equal to a fitted parameter vector obtained when fitting the
conditional model to a bootstrapped version of the data set.

Class structure
---------------

There are two main classes in the module:

* 'MICEData' wraps a Pandas dataframe, incorporating information about
  the imputation model for each variable with missing values. It can
  be used to produce multiply imputed data sets that are to be further
  processed or distributed to other researchers.  A number of plotting
  procedures are provided to visualize the imputation results and
  missing data patterns.  The `history_func` hook allows any features
  of interest of the imputed data sets to be saved for further
  analysis.

* 'MICE' takes both a 'MICEData' object and an analysis model
  specification.  It runs the multiple imputation, fits the analysis
  models, and combines the results to produce a `MICEResults` object.
  The summary method of this results object can be used to see the key
  estimands and inferential quantities..

Notes
-----

By default, to conserve memory 'MICEData' saves very little
information from one iteration to the next.  The data set passed by
the user is copied on entry, but then is over-written each time new
imputations are produced.  If using 'MICE', the fitted
analysis models and results are saved.  MICEData includes a
`history_callback` hook that allows arbitrary information from the
intermediate datasets to be saved for future use.

References
----------

JL Schafer: 'Multiple Imputation: A Primer', Stat Methods Med Res,
1999.

TE Raghunathan et al.: 'A Multivariate Technique for Multiply
Imputing Missing Values Using a Sequence of Regression Models', Survey
Methodology, 2001.

SAS Institute: 'Predictive Mean Matching Method for Monotone Missing
Data', SAS 9.2 User's Guide, 2014.

A Gelman et al.: 'Multiple Imputation with Diagnostics (mi) in R:
Opening Windows into the Black Box', Journal of Statistical Software,
2009.
"""

import pandas as pd
import numpy as np
import patsy
from statsmodels.base.model import LikelihoodModelResults
from statsmodels.regression.linear_model import OLS
from collections import defaultdict


_mice_data_example_1 = """
    >>> imp = mice.MICEData(data)
    >>> imp.set_imputer('x1', formula='x2 + np.square(x2) + x3')
    >>> for j in range(20):
    ...     imp.update_all()
    ...     imp.data.to_csv('data%02d.csv' % j)"""

_mice_data_example_2 = """
    >>> imp = mice.MICEData(data)
    >>> j = 0
    >>> for data in imp:
    ...     imp.data.to_csv('data%02d.csv' % j)
    ...     j += 1"""


class PatsyFormula(object):
    """
    A simple wrapper for a string to be interpreted as a Patsy formula.
    """
    def __init__(self, formula):
        self.formula = "0 + " + formula


[docs]class MICEData(object): __doc__ = """\ Wrap a data set to allow missing data handling with MICE. Parameters ---------- data : Pandas data frame The data set, whch is copied internally. perturbation_method : string The default perturbation method k_pmm : int The number of nearest neighbors to use during predictive mean matching. Can also be specified in `fit`. history_callback : function A function that is called after each complete imputation cycle. The return value is appended to `history`. The MICEData object is passed as the sole argument to `history_callback`. Examples -------- Draw 20 imputations from a data set called `data` and save them in separate files with filename pattern `dataXX.csv`. The variables other than `x1` are imputed using linear models fit with OLS, with mean structures containing main effects of all other variables in `data`. The variable named `x1` has a condtional mean structure that includes an additional term for x2^2. %(_mice_data_example_1)s Impute using default models, using the MICEData object as an iterator. %(_mice_data_example_2)s Notes ----- Allowed perturbation methods are 'gaussian' (the model parameters are set to a draw from the Gaussian approximation to the posterior distribution), and 'boot' (the model parameters are set to the estimated values obtained when fitting a bootstrapped version of the data set). `history_callback` can be implemented to have side effects such as saving the current imputed data set to disk. """ % {'_mice_data_example_1': _mice_data_example_1, '_mice_data_example_2': _mice_data_example_2} def __init__(self, data, perturbation_method='gaussian', k_pmm=20, history_callback=None): if data.columns.dtype != np.dtype('O'): msg = "MICEData data column names should be string type" raise ValueError(msg) self.regularized = dict() # Drop observations where all variables are missing. This # also has the effect of copying the data frame. self.data = data.dropna(how='all').reset_index(drop=True) self.history_callback = history_callback self.history = [] self.predict_kwds = {} # Assign the same perturbation method for all variables. # Can be overriden when calling 'set_imputer'. self.perturbation_method = defaultdict(lambda: perturbation_method) # Map from variable name to indices of observed/missing # values. self.ix_obs = {} self.ix_miss = {} for col in self.data.columns: ix_obs, ix_miss = self._split_indices(self.data[col]) self.ix_obs[col] = ix_obs self.ix_miss[col] = ix_miss # Most recent model instance and results instance for each variable. self.models = {} self.results = {} # Map from variable names to the conditional formula. self.conditional_formula = {} # Map from variable names to init/fit args of the conditional # models. self.init_kwds = defaultdict(lambda: dict()) self.fit_kwds = defaultdict(lambda: dict()) # Map from variable names to the model class. self.model_class = {} # Map from variable names to most recent params update. self.params = {} # Set default imputers. for vname in data.columns: self.set_imputer(vname) # The order in which variables are imputed in each cycle. # Impute variables with the fewest missing values first. vnames = list(data.columns) nmiss = [len(self.ix_miss[v]) for v in vnames] nmiss = np.asarray(nmiss) ii = np.argsort(nmiss) ii = ii[sum(nmiss == 0):] self._cycle_order = [vnames[i] for i in ii] self._initial_imputation() self.k_pmm = k_pmm
[docs] def next_sample(self): """ Returns the next imputed dataset in the imputation process. Returns ------- data : array-like An imputed dataset from the MICE chain. Notes ----- `MICEData` does not have a `skip` parameter. Consecutive values returned by `next_sample` are immediately consecutive in the imputation chain. The returned value is a reference to the data attribute of the class and should be copied before making any changes. """ self.update_all(1) return self.data
def _initial_imputation(self): """ Use a PMM-like procedure for initial imputed values. For each variable, missing values are imputed as the observed value that is closest to the mean over all observed values. """ for col in self.data.columns: di = self.data[col] - self.data[col].mean() di = np.abs(di) ix = di.idxmin() imp = self.data[col].loc[ix] self.data[col].fillna(imp, inplace=True) def _split_indices(self, vec): null = pd.isnull(vec) ix_obs = np.flatnonzero(~null) ix_miss = np.flatnonzero(null) if len(ix_obs) == 0: raise ValueError("variable to be imputed has no observed values") return ix_obs, ix_miss
[docs] def set_imputer(self, endog_name, formula=None, model_class=None, init_kwds=None, fit_kwds=None, predict_kwds=None, k_pmm=20, perturbation_method=None, regularized=False): """ Specify the imputation process for a single variable. Parameters ---------- endog_name : string Name of the variable to be imputed. formula : string Conditional formula for imputation. Defaults to a formula with main effects for all other variables in dataset. The formula should only include an expression for the mean structure, e.g. use 'x1 + x2' not 'x4 ~ x1 + x2'. model_class : statsmodels model Conditional model for imputation. Defaults to OLS. See below for more information. init_kwds : dit-like Keyword arguments passed to the model init method. fit_kwds : dict-like Keyword arguments passed to the model fit method. predict_kwds : dict-like Keyword arguments passed to the model predict method. k_pmm : int Determines number of neighboring observations from which to randomly sample when using predictive mean matching. perturbation_method : string Either 'gaussian' or 'bootstrap'. Determines the method for perturbing parameters in the imputation model. If None, uses the default specified at class initialization. regularized : dict If regularized[name]=True, `fit_regularized` rather than `fit` is called when fitting imputation models for this variable. When regularized[name]=True for any variable, pertrurbation_method must be set to boot. Notes ----- The model class must meet the following conditions: * A model must have a 'fit' method that returns an object. * The object returned from `fit` must have a `params` attribute that is an array-like object. * The object returned from `fit` must have a cov_params method that returns a square array-like object. * The model must have a `predict` method. """ if formula is None: main_effects = [x for x in self.data.columns if x != endog_name] fml = endog_name + " ~ " + " + ".join(main_effects) self.conditional_formula[endog_name] = fml else: fml = endog_name + " ~ " + formula self.conditional_formula[endog_name] = fml if model_class is None: self.model_class[endog_name] = OLS else: self.model_class[endog_name] = model_class if init_kwds is not None: self.init_kwds[endog_name] = init_kwds if fit_kwds is not None: self.fit_kwds[endog_name] = fit_kwds if predict_kwds is not None: self.predict_kwds[endog_name] = predict_kwds if perturbation_method is not None: self.perturbation_method[endog_name] = perturbation_method self.k_pmm = k_pmm self.regularized[endog_name] = regularized
def _store_changes(self, col, vals): """ Fill in dataset with imputed values. Parameters ---------- col : string Name of variable to be filled in. vals : array Array of imputed values to use for filling-in missing values. """ ix = self.ix_miss[col] if len(ix) > 0: self.data[col].iloc[ix] = np.atleast_1d(vals)
[docs] def update_all(self, n_iter=1): """ Perform a specified number of MICE iterations. Parameters ---------- n_iter : int The number of updates to perform. Only the result of the final update will be available. Notes ----- The imputed values are stored in the class attribute `self.data`. """ for k in range(n_iter): for vname in self._cycle_order: self.update(vname) if self.history_callback is not None: hv = self.history_callback(self) self.history.append(hv)
[docs] def get_split_data(self, vname): """ Return endog and exog for imputation of a given variable. Parameters ---------- vname : string The variable for which the split data is returned. Returns ------- endog_obs : DataFrame Observed values of the variable to be imputed. exog_obs : DataFrame Current values of the predictors where the variable to be imputed is observed. exog_miss : DataFrame Current values of the predictors where the variable to be Imputed is missing. init_kwds : dict-like The init keyword arguments for `vname`, processed through Patsy as required. fit_kwds : dict-like The fit keyword arguments for `vname`, processed through Patsy as required. """ formula = self.conditional_formula[vname] endog, exog = patsy.dmatrices(formula, self.data, return_type="dataframe") # Rows with observed endog ixo = self.ix_obs[vname] endog_obs = np.asarray(endog.iloc[ixo]) exog_obs = np.asarray(exog.iloc[ixo, :]) # Rows with missing endog ixm = self.ix_miss[vname] exog_miss = np.asarray(exog.iloc[ixm, :]) predict_obs_kwds = {} if vname in self.predict_kwds: kwds = self.predict_kwds[vname] predict_obs_kwds = self._process_kwds(kwds, ixo) predict_miss_kwds = {} if vname in self.predict_kwds: kwds = self.predict_kwds[vname] predict_miss_kwds = self._process_kwds(kwds, ixo) return (endog_obs, exog_obs, exog_miss, predict_obs_kwds, predict_miss_kwds)
def _process_kwds(self, kwds, ix): kwds = kwds.copy() for k in kwds: v = kwds[k] if isinstance(v, PatsyFormula): mat = patsy.dmatrix(v.formula, self.data, return_type="dataframe") mat = np.asarray(mat)[ix, :] if mat.shape[1] == 1: mat = mat[:, 0] kwds[k] = mat return kwds
[docs] def get_fitting_data(self, vname): """ Return the data needed to fit a model for imputation. The data is used to impute variable `vname`, and therefore only includes cases for which `vname` is observed. Values of type `PatsyFormula` in `init_kwds` or `fit_kwds` are processed through Patsy and subset to align with the model's endog and exog. Parameters ---------- vname : string The variable for which the fitting data is returned. Returns ------- endog : DataFrame Observed values of `vname`. exog : DataFrame Regression design matrix for imputing `vname`. init_kwds : dict-like The init keyword arguments for `vname`, processed through Patsy as required. fit_kwds : dict-like The fit keyword arguments for `vname`, processed through Patsy as required. """ # Rows with observed endog ix = self.ix_obs[vname] formula = self.conditional_formula[vname] endog, exog = patsy.dmatrices(formula, self.data, return_type="dataframe") endog = np.asarray(endog.iloc[ix, 0]) exog = np.asarray(exog.iloc[ix, :]) init_kwds = self._process_kwds(self.init_kwds[vname], ix) fit_kwds = self._process_kwds(self.fit_kwds[vname], ix) return endog, exog, init_kwds, fit_kwds
[docs] def plot_missing_pattern(self, ax=None, row_order="pattern", column_order="pattern", hide_complete_rows=False, hide_complete_columns=False, color_row_patterns=True): """ Generate an image showing the missing data pattern. Parameters ---------- ax : matplotlib axes Axes on which to draw the plot. row_order : string The method for ordering the rows. Must be one of 'pattern', 'proportion', or 'raw'. column_order : string The method for ordering the columns. Must be one of 'pattern', 'proportion', or 'raw'. hide_complete_rows : boolean If True, rows with no missing values are not drawn. hide_complete_columns : boolean If True, columns with no missing values are not drawn. color_row_patterns : boolean If True, color the unique row patterns, otherwise use grey and white as colors. Returns ------- A figure containing a plot of the missing data pattern. """ # Create an indicator matrix for missing values. miss = np.zeros(self.data.shape) cols = self.data.columns for j, col in enumerate(cols): ix = self.ix_miss[col] miss[ix, j] = 1 # Order the columns as requested if column_order == "proportion": ix = np.argsort(miss.mean(0)) elif column_order == "pattern": cv = np.cov(miss.T) u, s, vt = np.linalg.svd(cv, 0) ix = np.argsort(cv[:, 0]) elif column_order == "raw": ix = np.arange(len(cols)) else: raise ValueError( column_order + " is not an allowed value for `column_order`.") miss = miss[:, ix] cols = [cols[i] for i in ix] # Order the rows as requested if row_order == "proportion": ix = np.argsort(miss.mean(1)) elif row_order == "pattern": x = 2**np.arange(miss.shape[1]) rky = np.dot(miss, x) ix = np.argsort(rky) elif row_order == "raw": ix = np.arange(miss.shape[0]) else: raise ValueError( row_order + " is not an allowed value for `row_order`.") miss = miss[ix, :] if hide_complete_rows: ix = np.flatnonzero((miss == 1).any(1)) miss = miss[ix, :] if hide_complete_columns: ix = np.flatnonzero((miss == 1).any(0)) miss = miss[:, ix] cols = [cols[i] for i in ix] from statsmodels.graphics import utils as gutils from matplotlib.colors import LinearSegmentedColormap if ax is None: fig, ax = gutils.create_mpl_ax(ax) else: fig = ax.get_figure() if color_row_patterns: x = 2**np.arange(miss.shape[1]) rky = np.dot(miss, x) _, rcol = np.unique(rky, return_inverse=True) miss *= 1 + rcol[:, None] ax.imshow(miss, aspect="auto", interpolation="nearest", cmap='gist_ncar_r') else: cmap = LinearSegmentedColormap.from_list("_", ["white", "darkgrey"]) ax.imshow(miss, aspect="auto", interpolation="nearest", cmap=cmap) ax.set_ylabel("Cases") ax.set_xticks(range(len(cols))) ax.set_xticklabels(cols, rotation=90) return fig
[docs] def plot_bivariate(self, col1_name, col2_name, lowess_args=None, lowess_min_n=40, jitter=None, plot_points=True, ax=None): """ Plot observed and imputed values for two variables. Displays a scatterplot of one variable against another. The points are colored according to whether the values are observed or imputed. Parameters ---------- col1_name : string The variable to be plotted on the horizontal axis. col2_name : string The variable to be plotted on the vertical axis. lowess_args : dictionary A dictionary of dictionaries, keys are 'ii', 'io', 'oi' and 'oo', where 'o' denotes 'observed' and 'i' denotes imputed. See Notes for details. lowess_min_n : integer Minimum sample size to plot a lowess fit jitter : float or tuple Standard deviation for jittering points in the plot. Either a single scalar applied to both axes, or a tuple containing x-axis jitter and y-axis jitter, respectively. plot_points : bool If True, the data points are plotted. ax : matplotlib axes object Axes on which to plot, created if not provided. Returns ------- The matplotlib figure on which the plot id drawn. """ from statsmodels.graphics import utils as gutils from statsmodels.nonparametric.smoothers_lowess import lowess if lowess_args is None: lowess_args = {} if ax is None: fig, ax = gutils.create_mpl_ax(ax) else: fig = ax.get_figure() ax.set_position([0.1, 0.1, 0.7, 0.8]) ix1i = self.ix_miss[col1_name] ix1o = self.ix_obs[col1_name] ix2i = self.ix_miss[col2_name] ix2o = self.ix_obs[col2_name] ix_ii = np.intersect1d(ix1i, ix2i) ix_io = np.intersect1d(ix1i, ix2o) ix_oi = np.intersect1d(ix1o, ix2i) ix_oo = np.intersect1d(ix1o, ix2o) vec1 = np.asarray(self.data[col1_name]) vec2 = np.asarray(self.data[col2_name]) if jitter is not None: if np.isscalar(jitter): jitter = (jitter, jitter) vec1 += jitter[0] * np.random.normal(size=len(vec1)) vec2 += jitter[1] * np.random.normal(size=len(vec2)) # Plot the points keys = ['oo', 'io', 'oi', 'ii'] lak = {'i': 'imp', 'o': 'obs'} ixs = {'ii': ix_ii, 'io': ix_io, 'oi': ix_oi, 'oo': ix_oo} color = {'oo': 'grey', 'ii': 'red', 'io': 'orange', 'oi': 'lime'} if plot_points: for ky in keys: ix = ixs[ky] lab = lak[ky[0]] + "/" + lak[ky[1]] ax.plot(vec1[ix], vec2[ix], 'o', color=color[ky], label=lab, alpha=0.6) # Plot the lowess fits for ky in keys: ix = ixs[ky] if len(ix) < lowess_min_n: continue if ky in lowess_args: la = lowess_args[ky] else: la = {} ix = ixs[ky] lfit = lowess(vec2[ix], vec1[ix], **la) if plot_points: ax.plot(lfit[:, 0], lfit[:, 1], '-', color=color[ky], alpha=0.6, lw=4) else: lab = lak[ky[0]] + "/" + lak[ky[1]] ax.plot(lfit[:, 0], lfit[:, 1], '-', color=color[ky], alpha=0.6, lw=4, label=lab) ha, la = ax.get_legend_handles_labels() pad = 0.0001 if plot_points else 0.5 leg = fig.legend(ha, la, 'center right', numpoints=1, handletextpad=pad) leg.draw_frame(False) ax.set_xlabel(col1_name) ax.set_ylabel(col2_name) return fig
[docs] def plot_fit_obs(self, col_name, lowess_args=None, lowess_min_n=40, jitter=None, plot_points=True, ax=None): """ Plot fitted versus imputed or observed values as a scatterplot. Parameters ---------- col_name : string The variable to be plotted on the horizontal axis. lowess_args : dict-like Keyword arguments passed to lowess fit. A dictionary of dictionaries, keys are 'o' and 'i' denoting 'observed' and 'imputed', respectively. lowess_min_n : integer Minimum sample size to plot a lowess fit jitter : float or tuple Standard deviation for jittering points in the plot. Either a single scalar applied to both axes, or a tuple containing x-axis jitter and y-axis jitter, respectively. plot_points : bool If True, the data points are plotted. ax : matplotlib axes object Axes on which to plot, created if not provided. Returns ------- The matplotlib figure on which the plot is drawn. """ from statsmodels.graphics import utils as gutils from statsmodels.nonparametric.smoothers_lowess import lowess if lowess_args is None: lowess_args = {} if ax is None: fig, ax = gutils.create_mpl_ax(ax) else: fig = ax.get_figure() ax.set_position([0.1, 0.1, 0.7, 0.8]) ixi = self.ix_miss[col_name] ixo = self.ix_obs[col_name] vec1 = np.asarray(self.data[col_name]) # Fitted values formula = self.conditional_formula[col_name] endog, exog = patsy.dmatrices(formula, self.data, return_type="dataframe") results = self.results[col_name] vec2 = results.predict(exog=exog) vec2 = self._get_predicted(vec2) if jitter is not None: if np.isscalar(jitter): jitter = (jitter, jitter) vec1 += jitter[0] * np.random.normal(size=len(vec1)) vec2 += jitter[1] * np.random.normal(size=len(vec2)) # Plot the points keys = ['o', 'i'] ixs = {'o': ixo, 'i': ixi} lak = {'o': 'obs', 'i': 'imp'} color = {'o': 'orange', 'i': 'lime'} if plot_points: for ky in keys: ix = ixs[ky] ax.plot(vec1[ix], vec2[ix], 'o', color=color[ky], label=lak[ky], alpha=0.6) # Plot the lowess fits for ky in keys: ix = ixs[ky] if len(ix) < lowess_min_n: continue if ky in lowess_args: la = lowess_args[ky] else: la = {} ix = ixs[ky] lfit = lowess(vec2[ix], vec1[ix], **la) ax.plot(lfit[:, 0], lfit[:, 1], '-', color=color[ky], alpha=0.6, lw=4, label=lak[ky]) ha, la = ax.get_legend_handles_labels() leg = fig.legend(ha, la, 'center right', numpoints=1) leg.draw_frame(False) ax.set_xlabel(col_name + " observed or imputed") ax.set_ylabel(col_name + " fitted") return fig
[docs] def plot_imputed_hist(self, col_name, ax=None, imp_hist_args=None, obs_hist_args=None, all_hist_args=None): """ Display imputed values for one variable as a histogram. Parameters ---------- col_name : string The name of the variable to be plotted. ax : matplotlib axes An axes on which to draw the histograms. If not provided, one is created. imp_hist_args : dict Keyword arguments to be passed to pyplot.hist when creating the histogram for imputed values. obs_hist_args : dict Keyword arguments to be passed to pyplot.hist when creating the histogram for observed values. all_hist_args : dict Keyword arguments to be passed to pyplot.hist when creating the histogram for all values. Returns ------- The matplotlib figure on which the histograms were drawn """ from statsmodels.graphics import utils as gutils if imp_hist_args is None: imp_hist_args = {} if obs_hist_args is None: obs_hist_args = {} if all_hist_args is None: all_hist_args = {} if ax is None: fig, ax = gutils.create_mpl_ax(ax) else: fig = ax.get_figure() ax.set_position([0.1, 0.1, 0.7, 0.8]) ixm = self.ix_miss[col_name] ixo = self.ix_obs[col_name] imp = self.data[col_name].iloc[ixm] obs = self.data[col_name].iloc[ixo] for di in imp_hist_args, obs_hist_args, all_hist_args: if 'histtype' not in di: di['histtype'] = 'step' ha, la = [], [] if len(imp) > 0: h = ax.hist(np.asarray(imp), **imp_hist_args) ha.append(h[-1][0]) la.append("Imp") h1 = ax.hist(np.asarray(obs), **obs_hist_args) h2 = ax.hist(np.asarray(self.data[col_name]), **all_hist_args) ha.extend([h1[-1][0], h2[-1][0]]) la.extend(["Obs", "All"]) leg = fig.legend(ha, la, 'center right', numpoints=1) leg.draw_frame(False) ax.set_xlabel(col_name) ax.set_ylabel("Frequency") return fig
# Try to identify any auxiliary arrays (e.g. status vector in # PHReg) that need to be bootstrapped along with exog and endog. def _boot_kwds(self, kwds, rix): for k in kwds: v = kwds[k] # This is only relevant for ndarrays if not isinstance(v, np.ndarray): continue # Handle 1d vectors if (v.ndim == 1) and (v.shape[0] == len(rix)): kwds[k] = v[rix] # Handle 2d arrays if (v.ndim == 2) and (v.shape[0] == len(rix)): kwds[k] = v[rix, :] return kwds def _perturb_bootstrap(self, vname): """ Perturbs the model's parameters using a bootstrap. """ endog, exog, init_kwds, fit_kwds = self.get_fitting_data(vname) m = len(endog) rix = np.random.randint(0, m, m) endog = endog[rix] exog = exog[rix, :] init_kwds = self._boot_kwds(init_kwds, rix) fit_kwds = self._boot_kwds(fit_kwds, rix) klass = self.model_class[vname] self.models[vname] = klass(endog, exog, **init_kwds) if vname in self.regularized and self.regularized[vname]: self.results[vname] = ( self.models[vname].fit_regularized(**fit_kwds)) else: self.results[vname] = self.models[vname].fit(**fit_kwds) self.params[vname] = self.results[vname].params def _perturb_gaussian(self, vname): """ Gaussian perturbation of model parameters. The normal approximation to the sampling distribution of the parameter estimates is used to define the mean and covariance structure of the perturbation distribution. """ endog, exog, init_kwds, fit_kwds = self.get_fitting_data(vname) klass = self.model_class[vname] self.models[vname] = klass(endog, exog, **init_kwds) self.results[vname] = self.models[vname].fit(**fit_kwds) cov = self.results[vname].cov_params() mu = self.results[vname].params self.params[vname] = np.random.multivariate_normal(mean=mu, cov=cov)
[docs] def perturb_params(self, vname): if self.perturbation_method[vname] == "gaussian": self._perturb_gaussian(vname) elif self.perturbation_method[vname] == "boot": self._perturb_bootstrap(vname) else: raise ValueError("unknown perturbation method")
[docs] def impute(self, vname): # Wrap this in case we later add additional imputation # methods. self.impute_pmm(vname)
[docs] def update(self, vname): """ Impute missing values for a single variable. This is a two-step process in which first the parameters are perturbed, then the missing values are re-imputed. Parameters ---------- vname : string The name of the variable to be updated. """ self.perturb_params(vname) self.impute(vname)
# work-around for inconsistent predict return values def _get_predicted(self, obj): if isinstance(obj, np.ndarray): return obj elif isinstance(obj, pd.Series): return obj.values elif hasattr(obj, 'predicted_values'): return obj.predicted_values else: raise ValueError( "cannot obtain predicted values from %s" % obj.__class__)
[docs] def impute_pmm(self, vname): """ Use predictive mean matching to impute missing values. Notes ----- The `perturb_params` method must be called first to define the model. """ k_pmm = self.k_pmm endog_obs, exog_obs, exog_miss, predict_obs_kwds, predict_miss_kwds = ( self.get_split_data(vname)) # Predict imputed variable for both missing and non-missing # observations model = self.models[vname] pendog_obs = model.predict(self.params[vname], exog_obs, **predict_obs_kwds) pendog_miss = model.predict(self.params[vname], exog_miss, **predict_miss_kwds) pendog_obs = self._get_predicted(pendog_obs) pendog_miss = self._get_predicted(pendog_miss) # Jointly sort the observed and predicted endog values for the # cases with observed values. ii = np.argsort(pendog_obs) endog_obs = endog_obs[ii] pendog_obs = pendog_obs[ii] # Find the closest match to the predicted endog values for # cases with missing endog values. ix = np.searchsorted(pendog_obs, pendog_miss) # Get the indices for the closest k_pmm values on # either side of the closest index. ixm = ix[:, None] + np.arange(-k_pmm, k_pmm)[None, :] # Account for boundary effects msk = np.nonzero((ixm < 0) | (ixm > len(endog_obs) - 1)) ixm = np.clip(ixm, 0, len(endog_obs) - 1) # Get the distances dx = pendog_miss[:, None] - pendog_obs[ixm] dx = np.abs(dx) dx[msk] = np.inf # Closest positions in ix, row-wise. dxi = np.argsort(dx, 1)[:, 0:k_pmm] # Choose a column for each row. ir = np.random.randint(0, k_pmm, len(pendog_miss)) # Unwind the indices jj = np.arange(dxi.shape[0]) ix = dxi[(jj, ir)] iz = ixm[(jj, ix)] imputed_miss = np.array(endog_obs[iz]).squeeze() self._store_changes(vname, imputed_miss)
_mice_example_1 = """ >>> imp = mice.MICEData(data) >>> fml = 'y ~ x1 + x2 + x3 + x4' >>> mice = mice.MICE(fml, sm.OLS, imp) >>> results = mice.fit(10, 10) >>> print(results.summary()) :: Results: MICE ================================================================= Method: MICE Sample size: 1000 Model: OLS Scale 1.00 Dependent variable: y Num. imputations 10 ----------------------------------------------------------------- Coef. Std.Err. t P>|t| [0.025 0.975] FMI ----------------------------------------------------------------- Intercept -0.0234 0.0318 -0.7345 0.4626 -0.0858 0.0390 0.0128 x1 1.0305 0.0578 17.8342 0.0000 0.9172 1.1437 0.0309 x2 -0.0134 0.0162 -0.8282 0.4076 -0.0451 0.0183 0.0236 x3 -1.0260 0.0328 -31.2706 0.0000 -1.0903 -0.9617 0.0169 x4 -0.0253 0.0336 -0.7520 0.4521 -0.0911 0.0406 0.0269 ================================================================= """ _mice_example_2 = """ >>> imp = mice.MICEData(data) >>> fml = 'y ~ x1 + x2 + x3 + x4' >>> mice = mice.MICE(fml, sm.OLS, imp) >>> results = [] >>> for k in range(10): >>> x = mice.next_sample() >>> results.append(x) """
[docs]class MICE(object): __doc__ = """\ Multiple Imputation with Chained Equations. This class can be used to fit most Statsmodels models to data sets with missing values using the 'multiple imputation with chained equations' (MICE) approach.. Parameters ---------- model_formula : string The model formula to be fit to the imputed data sets. This formula is for the 'analysis model'. model_class : statsmodels model The model to be fit to the imputed data sets. This model class if for the 'analysis model'. data : MICEData instance MICEData object containing the data set for which missing values will be imputed n_skip : int The number of imputed datasets to skip between consecutive imputed datasets that are used for analysis. init_kwds : dict-like Dictionary of keyword arguments passed to the init method of the analysis model. fit_kwds : dict-like Dictionary of keyword arguments passed to the fit method of the analysis model. Examples -------- Run all MICE steps and obtain results: %(mice_example_1)s Obtain a sequence of fitted analysis models without combining to obtain summary: %(mice_example_2)s """ % {'mice_example_1': _mice_example_1, 'mice_example_2': _mice_example_2} def __init__(self, model_formula, model_class, data, n_skip=3, init_kwds=None, fit_kwds=None): self.model_formula = model_formula self.model_class = model_class self.n_skip = n_skip self.data = data self.results_list = [] self.init_kwds = init_kwds if init_kwds is not None else {} self.fit_kwds = fit_kwds if fit_kwds is not None else {}
[docs] def next_sample(self): """ Perform one complete MICE iteration. A single MICE iteration updates all missing values using their respective imputation models, then fits the analysis model to the imputed data. Returns ------- params : array-like The model parameters for the analysis model. Notes ----- This function fits the analysis model and returns its parameter estimate. The parameter vector is not stored by the class and is not used in any subsequent calls to `combine`. Use `fit` to run all MICE steps together and obtain summary results. The complete cycle of missing value imputation followed by fitting the analysis model is repeated `n_skip + 1` times and the analysis model parameters from the final fit are returned. """ # Impute missing values self.data.update_all(self.n_skip + 1) start_params = None if len(self.results_list) > 0: start_params = self.results_list[-1].params # Fit the analysis model. model = self.model_class.from_formula(self.model_formula, self.data.data, **self.init_kwds) self.fit_kwds.update({"start_params": start_params}) result = model.fit(**self.fit_kwds) return result
[docs] def fit(self, n_burnin=10, n_imputations=10): """ Fit a model using MICE. Parameters ---------- n_burnin : int The number of burn-in cycles to skip. n_imputations : int The number of data sets to impute """ # Run without fitting the analysis model self.data.update_all(n_burnin) for j in range(n_imputations): result = self.next_sample() self.results_list.append(result) self.endog_names = result.model.endog_names self.exog_names = result.model.exog_names return self.combine()
[docs] def combine(self): """ Pools MICE imputation results. This method can only be used after the `run` method has been called. Returns estimates and standard errors of the analysis model parameters. Returns a MICEResults instance. """ # Extract a few things from the models that were fit to # imputed data sets. params_list = [] cov_within = 0. scale_list = [] for results in self.results_list: results_uw = results._results params_list.append(results_uw.params) cov_within += results_uw.cov_params() scale_list.append(results.scale) params_list = np.asarray(params_list) scale_list = np.asarray(scale_list) # The estimated parameters for the MICE analysis params = params_list.mean(0) # The average of the within-imputation covariances cov_within /= len(self.results_list) # The between-imputation covariance cov_between = np.cov(params_list.T) # The estimated covariance matrix for the MICE analysis f = 1 + 1 / float(len(self.results_list)) cov_params = cov_within + f * cov_between # Fraction of missing information fmi = f * np.diag(cov_between) / np.diag(cov_params) # Set up a results instance scale = np.mean(scale_list) results = MICEResults(self, params, cov_params / scale) results.scale = scale results.frac_miss_info = fmi results.exog_names = self.exog_names results.endog_names = self.endog_names results.model_class = self.model_class return results
class MICEResults(LikelihoodModelResults): def __init__(self, model, params, normalized_cov_params): super(MICEResults, self).__init__(model, params, normalized_cov_params) def summary(self, title=None, alpha=.05): """ Summarize the results of running MICE. Parameters ----------- title : string, optional Title for the top table. If not None, then this replaces the default title alpha : float Significance level for the confidence intervals Returns ------- smry : Summary instance This holds the summary tables and text, which can be printed or converted to various output formats. """ from statsmodels.iolib import summary2 from collections import OrderedDict smry = summary2.Summary() float_format = "%8.3f" info = OrderedDict() info["Method:"] = "MICE" info["Model:"] = self.model_class.__name__ info["Dependent variable:"] = self.endog_names info["Sample size:"] = "%d" % self.model.data.data.shape[0] info["Scale"] = "%.2f" % self.scale info["Num. imputations"] = "%d" % len(self.model.results_list) smry.add_dict(info, align='l', float_format=float_format) param = summary2.summary_params(self, alpha=alpha) param["FMI"] = self.frac_miss_info smry.add_df(param, float_format=float_format) smry.add_title(title=title, results=self) return smry