M/PhVdZddlmZddlmZddlZddlmZddl Z ddl Z ddl m Z ddlmZddlmcmZddlmcmZddlmcmZddlmZdd lmZmZdd lmZ ddl!m"cmcm#Z#dd l$m%Z%dd l&m'Z'm(Z(m)Z)m*Z*ddl+Z+dd l,m-Z-m.Z.m/Z/ddl0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9GddZ:dZ;dZdZ?dZ@dZAdZBdZCdZDdZEGddeZFGddeZGGd d!ejHZIejJeIeGGd"d#eFZKGd$d%eGZLd&ZMGd'd(eIZNejJeNeLGd)d*eFZOGd+d,eGZPGd-d.eIZQejJeQePGd/d0e%ZRGd1d2ejSZTGd3d4ZUdS)5a` Procedures for fitting marginal regression models to dependent data using Generalized Estimating Equations. References ---------- KY Liang and S Zeger. "Longitudinal data analysis using generalized linear models". Biometrika (1986) 73 (1): 13-22. S Zeger and KY Liang. "Longitudinal Data Analysis for Discrete and Continuous Outcomes". Biometrics Vol. 42, No. 1 (Mar., 1986), pp. 121-130 A Rotnitzky and NP Jewell (1990). "Hypothesis testing of regression parameters in semiparametric generalized linear models for cluster correlated data", Biometrika, 77, 485-497. Xu Guo and Wei Pan (2002). "Small sample performance of the score test in GEE". http://www.sph.umn.edu/faculty1/wp-content/uploads/2012/11/rr2002-013.pdf LA Mancl LA, TA DeRouen (2001). A covariance estimator for GEE with improved small-sample properties. Biometrics. 2001 Mar;57(1):126-34. )lzip)AppenderN)stats) defaultdict)cache_readonly)families)GLM GLMResults) cov_struct)Link)ConvergenceWarning DomainWarningIterationLimitWarning ValueWarning)_plot_added_variable_doc_plot_partial_residuals_doc_plot_ceres_residuals_doc) _get_margeff_exog_check_margeff_args _effects_atmargeff_cov_with_se_check_at_is_all_transform_names_check_discrete_args_get_dummy_index_get_count_indexc6eZdZdZdZdZdZdZdZdZ dS) ParameterConstraintzV A class for managing linear equality constraints for a parameter vector. ctj|}|jdkrt dt ||jdkrt d||_||_tj |j d\}}}|ddt |df|_ |dddt |f|_ tj|j |j f|_tj|j tj||j|z |_tj||j|_||_tj||j|_dS)a Parameters ---------- lhs : ndarray A q x p matrix which is the left hand side of the constraint lhs * param = rhs. The number of constraints is q >= 1 and p is the dimension of the parameter vector. rhs : ndarray A 1-dimensional vector of length q which is the right hand side of the constraint equation. exog : ndarray The n x p exognenous data for the full model. z7The right hand side of the constraint must be a vector.rz~The number of rows of the left hand side constraint matrix L must equal the length of the right hand side constraint vector R.) full_matricesN)np atleast_1dsqueezendim ValueErrorlenshapelhsrhslinalgsvdTlhs0lhs1hstacklhsfdotparam0_offset_increment orig_exogexog_fulltrans)selfr)r*exoglhs_ulhs_slhs_vts s/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/statsmodels/genmod/generalized_estimating_equations.py__init__zParameterConstraint.__init__Dsa"mCKKMM** 8a<<122 2 s88sy| # #455 5  "y}}SU!}DDuf!!!SZZ[[.) !!!Qs5zz\/* Ity$)455 fTYvtx(@(@")#$$ "$dk!:!: fT4955c|jS)z Returns a vector that should be added to the offset vector to accommodate the constraint. Parameters ---------- exog : array_like The exogeneous data for the model. )r4r7s r<offset_incrementz$ParameterConstraint.offset_incrementws %%r>cH|jddd|jjdfS)z Returns a linearly transformed exog matrix whose columns span the constrained model space. Parameters ---------- exog : array_like The exogeneous data for the model. Nrr )r6r.r(r@s r< reduced_exogz ParameterConstraint.reduced_exogs)"111a (:&:#:;;r>c|jS)zg Returns the full exog matrix before it was reduced to satisfy the constraint. )r5r@s r< restore_exogz ParameterConstraint.restore_exogs ~r>cF|jtj|j|zS)zb Converts the parameter vector `params` from reduced to full coordinates. )r3r"r2r.r7paramss r< unpack_paramz ParameterConstraint.unpack_params {RVDIv6666r>cptj|jtj||jjS)za Converts the covariance matrix `bcov` from reduced to full coordinates. )r"r2r.r-)r7bcovs r< unpack_covzParameterConstraint.unpack_covs( vdidik!:!:;;;r>N) __name__ __module__ __qualname____doc__r=rArCrErIrLr>r<rr>sy 161616f & & & < < <777<<<<ra6 Marginal regression model fit using Generalized Estimating Equations. GEE can be used to fit Generalized Linear Models (GLMs) when the data have a grouped structure, and the observations are possibly correlated within groups but not between groups. Parameters ---------- endog : array_like 1d array of endogenous values (i.e. responses, outcomes, dependent variables, or 'Y' values). exog : array_like 2d array of exogeneous values (i.e. covariates, predictors, independent variables, regressors, or 'X' values). A `nobs x k` array where `nobs` is the number of observations and `k` is the number of regressors. An intercept is not included by default and should be added by the user. See `statsmodels.tools.add_constant`. groups : array_like A 1d array of length `nobs` containing the group labels. time : array_like A 2d array of time (or other index) values, used by some dependence structures to define similarity relationships among observations within a cluster. family : family class instance %(family_doc)s cov_struct : CovStruct class instance The default is Independence. To specify an exchangeable structure use cov_struct = Exchangeable(). See statsmodels.genmod.cov_struct.CovStruct for more information. offset : array_like An offset to be included in the fit. If provided, must be an array whose length is the number of rows in exog. dep_data : array_like Additional data passed to the dependence structure. constraint : (ndarray, ndarray) If provided, the constraint is a tuple (L, R) such that the model parameters are estimated under the constraint L * param = R, where L is a q x p matrix and R is a q-dimensional vector. If constraint is provided, a score test is performed to compare the constrained model to the unconstrained model. update_dep : bool If true, the dependence parameters are optimized, otherwise they are held fixed at their starting values. weights : array_like An array of case weights to use in the analysis. %(extra_params)s See Also -------- statsmodels.genmod.families.family :ref:`families` :ref:`links` Notes ----- Only the following combinations make sense for family and link :: + ident log logit probit cloglog pow opow nbinom loglog logc Gaussian | x x x inv Gaussian | x x x binomial | x x x x x x x x x Poisson | x x x neg binomial | x x x x gamma | x x x Not all of these link functions are currently available. Endog and exog are references so that if the data they refer to are already arrays and these arrays are changed, endog and exog will change. The "robust" covariance type is the standard "sandwich estimator" (e.g. Liang and Zeger (1986)). It is the default here and in most other packages. The "naive" estimator gives smaller standard errors, but is only correct if the working correlation structure is correctly specified. The "bias reduced" estimator of Mancl and DeRouen (Biometrics, 2001) reduces the downward bias of the robust estimator. The robust covariance provided here follows Liang and Zeger (1986) and agrees with R's gee implementation. To obtain the robust standard errors reported in Stata, multiply by sqrt(N / (N - g)), where N is the total sample size, and g is the average group size. %(notes)s Examples -------- %(example)s z The nominal and ordinal GEE models should not have an intercept (either implicit or explicit). Use "0 + " in a formula to suppress the intercept. z The default is Gaussian. To specify the binomial distribution use `family=sm.families.Binomial()`. Each family can take a link instance as an argument. See statsmodels.genmod.families.family for more information.z| The only family supported is `Binomial`. The default `Logit` link may be replaced with `probit` if desired.z The default value `None` uses a multinomial logit family specifically designed for use with GEE. Setting this argument to a non-default value is not currently supported.a Fits a marginal regression model using generalized estimating equations (GEE). Parameters ---------- maxiter : int The maximum number of iterations ctol : float The convergence criterion for stopping the Gauss-Seidel iterations start_params : array_like A vector of starting values for the regression coefficients. If None, a default is chosen. params_niter : int The number of Gauss-Seidel updates of the mean structure parameters that take place prior to each update of the dependence structure. first_dep_update : int No dependence structure updates occur before this iteration number. cov_type : str One of "robust", "naive", or "bias_reduced". ddof_scale : scalar or None The scale parameter is estimated as the sum of squared Pearson residuals divided by `N - ddof_scale`, where N is the total sample size. If `ddof_scale` is None, the number of covariates (including an intercept if present) is used. scaling_factor : scalar The estimated covariance of the parameter estimates is scaled by this value. Default is 1, Stata uses N / (N - g), where N is the total sample size and g is the average group size. scale : str or float, optional `scale` can be None, 'X2', or a float If a float, its value is used as the scale parameter. The default value is None, which uses `X2` (Pearson's chi-square) for Gamma, Gaussian, and Inverse Gaussian. The default is 1 for the Binomial and Poisson families. Returns ------- An instance of the GEEResults class or subclass Notes ----- If convergence difficulties occur, increase the values of `first_dep_update` and/or `params_niter`. Setting `first_dep_update` to a greater value (e.g. ~10-20) causes the algorithm to move close to the GLM solution before attempting to identify the dependence structure. For the Gaussian family, there is no benefit to setting `params_niter` to a value greater than 1, since the mean structure parameters converge in one step. aE Attributes ---------- cov_params_default : ndarray default covariance of the parameter estimates. Is chosen among one of the following three based on `cov_type` cov_robust : ndarray covariance of the parameter estimates that is robust cov_naive : ndarray covariance of the parameter estimates that is not robust to correlation or variance misspecification cov_robust_bc : ndarray covariance of the parameter estimates that is robust and bias reduced converged : bool indicator for convergence of the optimization. True if the norm of the score is smaller than a threshold cov_type : str string indicating whether a "robust", "naive" or "bias_reduced" covariance is used as default fit_history : dict Contains information about the iterations. fittedvalues : ndarray Linear predicted values for the fitted model. dot(exog, params) model : class instance Pointer to GEE model instance that called `fit`. normalized_cov_params : ndarray See GEE docstring params : ndarray The coefficients of the fitted model. Note that interpretation of the coefficients often depends on the distribution family and the data. scale : float The estimate of the scale / dispersion for the model fit. See GEE.fit for more information. score_norm : float norm of the score at the end of the iterative estimation. bse : ndarray The standard errors of the fitted GEE parameters. aF Logistic regression with autoregressive working dependence: >>> import statsmodels.api as sm >>> family = sm.families.Binomial() >>> va = sm.cov_struct.Autoregressive() >>> model = sm.GEE(endog, exog, group, family=family, cov_struct=va) >>> result = model.fit() >>> print(result.summary()) Use formulas to fit a Poisson GLM with independent working dependence: >>> import statsmodels.api as sm >>> fam = sm.families.Poisson() >>> ind = sm.cov_struct.Independence() >>> model = sm.GEE.from_formula("y ~ age + trt + base", "subject", data, cov_struct=ind, family=fam) >>> result = model.fit() >>> print(result.summary()) Equivalent, using the formula API: >>> import statsmodels.api as sm >>> import statsmodels.formula.api as smf >>> fam = sm.families.Poisson() >>> ind = sm.cov_struct.Independence() >>> model = smf.gee("y ~ age + trt + base", "subject", data, cov_struct=ind, family=fam) >>> result = model.fit() >>> print(result.summary()) a, Fit an ordinal regression model using GEE, with "global odds ratio" dependence: >>> import statsmodels.api as sm >>> gor = sm.cov_struct.GlobalOddsRatio("ordinal") >>> model = sm.OrdinalGEE(endog, exog, groups, cov_struct=gor) >>> result = model.fit() >>> print(result.summary()) Using formulas: >>> import statsmodels.formula.api as smf >>> model = smf.ordinal_gee("y ~ 0 + x1 + x2", groups, data, cov_struct=gor) >>> result = model.fit() >>> print(result.summary()) a Fit a nominal regression model using GEE: >>> import statsmodels.api as sm >>> import statsmodels.formula.api as smf >>> gor = sm.cov_struct.GlobalOddsRatio("nominal") >>> model = sm.NominalGEE(endog, exog, groups, cov_struct=gor) >>> result = model.fit() >>> print(result.summary()) Using formulas: >>> import statsmodels.api as sm >>> model = sm.NominalGEE.from_formula("y ~ 0 + x1 + x2", groups, data, cov_struct=gor) >>> result = model.fit() >>> print(result.summary()) Using the formula API: >>> import statsmodels.formula.api as smf >>> model = smf.nominal_gee("y ~ 0 + x1 + x2", groups, data, cov_struct=gor) >>> result = model.fit() >>> print(result.summary()) cX|j|jdkrtd|j|jkrtd||j|jkrtd||j|jkrtd||j|jkrtddSdS)Nrz:Leading dimension of 'exog' should match length of 'endog'z.'groups' and 'endog' should have the same sizez,'time' and 'endog' should have the same sizez-'offset and 'endog' should have the same sizez0'exposure' and 'endog' should have the same size)sizer(r&)endogr8groupstimeoffsetexposures r< _check_argsrYs zTZ]""-.. .{ej  IJJJ TY%*44GHHH v{ej88HIII%*!ceZdZdeejeeddzzZdZ d(fd Z e d)fd Z d Z d Zd Zd Zd*d ZdZdZdZdZdZee d+dZdZdZ d,d!Zd"Zd#Z d-d%Zd.d'Z xZ!S)/GEEzF Marginal Regression Model using Generalized Estimating Equations.  extra_params family_docexamplenotesNnoneTc t|tur|||pt|jt |jsId}tj| |jj j |j j ttj|}d|vr:|d2|d}||}|||}|||}| | |} |d=||_| |_| |_| |_t't(|_t-j||f|||| | | ||d|t1|j|j|j|jt;|ddt;|dd|jgd |j d|j dn#tB$rYnwxYw|tEj#}n.tI|j tDj%stCd ||_&|tOj(}n.tI|j tNj)stCd ||_*d|_| tW| d krtCd | d j,d|jj,dkrtCdt[| d | d|j|_|j.(|xj.|j/z c_.n0|j/0|_.|j1|_tj2|jd\}}tgj4tj5tW|d}|6|jfdto|D}tq||_9||_:|;|j|_<|;|j|_=|j>|;|j>|_?tW|j<|_@|jF|jjAdkr|jdddf|_|;|j|_Bn4d|j<D|_BtjC|jB|_|j.$tjD|j.r|j.dkrd|_En|;|j.|_E| )|;|jjF|j_G||_&|j*H|d|j<D}t||_J|jj,ddz |_K|jJ|jj,dz |_Ltd|j<D}|dkr d|_dSdS)NzDThe {0} link function does not respect the domain of the {1} family. missing_idx)rUrVrWrXweightsdep_datamissingfamilyrWrX) update_dep constraintrhr freq_weights var_weightsz.GEE: `family` must be a genmod family instancez6GEE: `cov_struct` must be a genmod cov_struct instancez$GEE: `constraint` must be a 2-tuple.rr zbGEE: the left hand side of the constraint must have the same number of columns as the exog matrix.T)return_inverseint)indexdtypecNg|]!\}}|tj|f"SrQ)r"asarray).0klbgbs r< z GEE.__init__..\s0 K K K%!Rr2:be$$% K K Kr>c|g|]9}tjt|tjdddf:S)rqN)r"aranger'float64rtys r<rxz GEE.__init__..qsL)))3q66444QQQW=)))r>c,g|]}t|SrQr'r}s r<rxz GEE.__init__..s222qCFF222r>c,g|]}t|SrQrrtxs r<rxz GEE.__init__..s6661A666r>F)Ntyper[ _check_kwargs isinstancelinktuple safe_linkswarningswarnformat __class__rMrr"rsrgrfrjrirlist _fit_historysuperr=rYrTr8rUrVgetattr _init_keysextendremover&rGaussian issubclassFamilyrh cov_structs Independence CovStructr r'r(r_offset_exposurerAcopyrCuniquepdSeriesr{groupby enumeratedict group_indices group_labels cluster_listendog_liexog_lire weights_li num_groupr%time_li concatenateisscalar offset_lir6exog_fulltrans_li initializesumnobsdf_modeldf_residmax)r7rTr8rUrVrhr rgrWrXrfrjrirekwargsmsgiirixsedkgroup_nsmaxgrouprwrs @r<r=z GEE.__init__s` ::     v & & &  fk51B+C+CDD -3 cjj)>)G)/)9)BDD+---F## F " "vm'<'H''BBZFBx!##B<}%   $$'--  2V"6"*G"*G &  2 2+1  2 2 2  J I K I D(D ) ) D*d + +      . . . / / /  O " "> 2 2 2 O " "= 1 1 1 1    D  >&((FFf.@@ 4 "3444   $133JJj2K4IJJ 8 "7888%  !:!## !GHHH!}"1%);;; EFFF2*Q-2!#H!5!5F:  v  H%Fx $$W6 '+ 664,2,2.6,2 66 /5 66  %$2E !( r>cljdkrfdjDSfdjDS)zf Returns `array` split into subarrays corresponding to the cluster structure. r cZg|]'}tjj|(SrQr"arrayrrtrurr7s r<rxz$GEE.cluster_list.. s@000HU4#5a#89::000r>cbg|]+}tjj|ddf,SNrrs r<rxz$GEE.cluster_list..sI000HU4#5a#8!!!#;<==000r>)r%r)r7rs``r<rzGEE.cluster_listsw :??00000!.000 000000!.000 0r>c> |jj|_|j}|jjd|jjdkrd}t ||jjd|jjdkrd}t j|t|jt|jsd}t j|t|j t|j st jdtj |j |j sd}t j|t||j\}}|d }t |tj||j}|j }ddl}||j} |j |_ |||\} } | d }t j|t.dSt1|d s|jjd|_t1|d sd|_|\} } } tj|j| } tj| }n9#tjj$r"tj | }YnwxYwtj|jtj| |}tj|jtj| |}tj|jtj| |}tj|jtj||}tj|jtj||} tj!||}nL#tjj$r5tjtj ||}YnwxYw|tj|j|z } tj!||}nL#tjj$r5tjtj ||}YnwxYw|tj|j|z} tj!||}nL#tjj$r5tjtj ||}YnwxYw tj!||}nL#tjj$r5tjtj ||}YnwxYw|tj|jtj||z }||_ | |_dd l"m#} tj!||}nL#tjj$r5tjtj ||}YnwxYwtj||}tI|}d|%||z }|||dS)a Perform a score test for the given submodel against this model. Parameters ---------- submodel : GEEResults instance A fitted GEE model that is a submodel of this model. Returns ------- A dictionary with keys "statistic", "p-value", and "df", containing the score test statistic, its chi^2 p-value, and the degrees of freedom used to compute the p-value. Notes ----- The score test can be performed without calling 'fit' on the larger model. The provided submodel must be obtained from a fitted GEE. This method performs the same score test as can be obtained by fitting the GEE with a linear constraint and calling `score_test` on the results. References ---------- Xu Guo and Wei Pan (2002). "Small sample performance of the score test in GEE". http://www.sph.umn.edu/faculty1/wp-content/uploads/2012/11/rr2002-013.pdf rz3Model and submodel have different numbers of cases.r z4Model and submodel have the same number of variablesz/Model and submodel have different GLM families.z>%((DDD )&rvdB//00&rvdB//00&rvdB//00&rvdB//00&rvdB//00 :'22BBy$ : : : w//99BBB :bfWY333  :'22BBy$ : : : w//99BBB : RVGIr***  :'22BBy$ : : : w//99BBB : :'22BBy$ : : : w//99BBB : RVGIrvb"~~666 *-222222 <)//)V44CCy$ < < <& 22F;;CCC <&--v;;488OX>>> ,')) )sq>J3KK9 OAP#"P# Q%%AR.-R. S00AT98T9= UAV'&V'/ XAYYc:|j?t|jtjtjtjtfrdSn3t|jtrtj |jS|j }|j }|j }|jj}d}d}t|jD]}t#||dkr||\}} tj||} |||z | z } |jB|j|} |tj| | dzzz }|| z }|tj| dzz }|t#| z }||||jz zt|z z}|S)z0 Estimate the dispersion/scale. N?rrrm)rrrhrBinomialPoissonNegativeBinomial _Multinomialfloatr"rrrrvariancerangerr'sqrtrerrr) r7rTrrvarfuncscalefsumiexpvalrsdevresidfs r<estimate_scalezGEE.estimate_scales > !$+(98;K(0(A(4(677 r  . . ,8DN++ + ( y+&t~&& # #A58}}!!$QIFA7776??++D1X&$.E|'OA&UaZ 0111 +++E " $$01E$KK?@ r>cb|jj|}||dddfz}|S)a> Derivative of the expected endog with respect to the parameters. Parameters ---------- exog : array_like The exogeneous data at which the derivative is computed. lin_pred : array_like The values of the linear predictor. Returns ------- The value of the derivative of the expected endog with respect to the parameter vector. Notes ----- If there is an offset or exposure, it should be added to `lin_pred` prior to calling this function. N)rhr inverse_deriv)r7r8lin_predidldmats r< mean_derivzGEE.mean_derivs7,k,,X66c!!!T'l" r>ctj||}|||z }|jj|}tj||}|S)a  Derivative of the expected endog with respect to exog. Parameters ---------- exog : array_like Values of the independent variables at which the derivative is calculated. params : array_like Parameter values at which the derivative is calculated. offset_exposure : array_like, optional Combined offset and exposure. Returns ------- The derivative of the expected endog with respect to exog. )r"r2rhrr$outer)r7r8rHoffset_exposurer%r&r's r<mean_deriv_exogzGEE.mean_deriv_exogsS&6$''  &  'Hk,,X66xV$$ r>c|j}|j}t|dd}|j}|jj}d\}}t |jD]}||\} } ||| z } |||| } tj || } |||}| |z}| |dddfz}n| }| }|j | || ||f}|dSt|\}}|tj| j|z }|tj| j|z } tj||}nL#tjj$r5tjtj||}YnwxYw|jd|j j||fS)a Returns ------- update : array_like The update vector such that params + update is the next iterate when solving the score equations. score : array_like The current value of the score equations, not incorporating the scale parameter. If desired, multiply this vector by the scale parameter to incorporate the scale. rNrrNN cov_adjust)rrrrrhrrrr(r"rr covariance_matrix_solverr2r-r+rrrrappendr0)r7rTr8rerrbmatrrrlprr r'rwwresidwdmatrsltvinv_d vinv_residupdates r<rzGEE._update_mean_paramss |$ d33( +& et~&& 0 0A&q/KFC!Hv%E??47C00D7776??++D"AJqDz)?::AteV_66D|!zz!&t FJ BF466** *D RVDFJ// /EE 9Y__T511FFy$ 9 9 9VBINN400%88FFF 9 ,'.. O & ( ( (u}s2 EAFFc`|j}|j}|j}|jjj}g|_t|jD]k}t||dkrtj |||}| |||z }||}|j ||fldS)z cached_means should always contain the most recent calculation of the group-wise mean vectors. This function should be called every time the regression parameters are changed, to keep the cached means up to date. rN) rrrrhrinverserrrr'r"r2r2) r7 mean_paramsrTr8rWlinkinvrr4rs r<rzGEE.update_cached_means8s |+"*t~&& 4 4A58}}!!&a+..C!vay WS\\F   $ $fc] 3 3 3 3 4 4r>c|j}|j}t|dd}|jj}|j}d\}}t |jD]}||\} } ||| z } |||| } tj || } |||}| |z}| |dddfz}n| }| }|j | || ||f}|dSt|\}}|tj| j|z }tj| j|}|tj||z }|} tj|}n9#tjj$r"tj|}YnwxYw||z}tj|tj||}||jz}||jz}|||fS)a Returns the sampling covariance matrix of the regression parameters and related quantities. Returns ------- cov_robust : array_like The robust, or sandwich estimate of the covariance, which is meaningful even if the working covariance structure is incorrectly specified. cov_naive : array_like The model-based estimate of the covariance, which is meaningful if the covariance structure is correctly specified. cmat : array_like The center matrix of the sandwich expression, used in obtaining score test results. rNr.NNNN)rrrrhrrrrr(r"rr r1rr2r-r*r"r+rrrr)r7rTr8rerrr3rrrr4r r'rr5r6r7r8r9r: dvinv_residrbmati cov_naive cov_robusts r<rz GEE._covmatTs( |$ d33+&(  dt~&& 7 7A&q/KFC!Hv%E??47C00D7776??++D"AJqDz)?::4%22D|---!&t FJ BF466** *D&44K BH[+66 6DD##%% )IMM$''EEy$ ) ) )INN4((EEE )EM VE26$#6#677 T(( d)) 9d**sE;;3F10F1c||jz }|j}|j}|jj}|j}|}d}t|jD]q}||\} } ||| z } | ||| } tj || } |j | || | f}|dS|d}||z}tj||}tj|| jj}|j |j|nd}tjtjt+| |z | }|j | || |f}|dS|d}|tj| j|z|z }|tj||z }stj|tj||}||jz}|S)Nrr)rrrrhrrr"rrr(r"rr r1r2r-rerr+reyer'r*)r7rDrTr8rrrbcmrrr4r r'rr8r9hmatr!aresidsrt cov_robust_bcs r< _bc_covmatzGEE._bc_covmats 33  |+&( ##%%t~&& & &A&q/KFC!Hv%E??47C00D7776??++D?::4$**D|tt!WF eOF6&),,D6$'')D&*l&>""BAY__RVCJJ%7%7$%>FFF?::4&,,D|ttq'CbfTVS)))E1C 28C%% %CCy"&i*@*@AA ,, r>ctj|jrd}n|j}t|j|j|j||j}|}|j S)N)rhrWrk) r"rrr rTr8rhrefitrH)r7rWrresults r<_starting_paramszGEE._starting_paramssb ;t, - - +FF*FDJ $+! >>>}r><ư>r rrobustrc X| |_||jjd|_n|dkst d||_||_t t|_|j |dkrt d|| } n(tj |}| } || d} d} t|D]?} |\}}|t#jdt&n| |z } || tjtj|dz} |jd | |jd ||jd |jj| |kr| dks |jd urn2|jr)| |zdkr | |kr|| | dz } A| |krt#jd t6| t#jdt&dS|\}}}|t#jdt&dSd}|dkr||}|j| }|| |\} }| t#jdt&dS|||\}}|t#jdt&dS|7|||\}}|t#jdt&dS| } tC||||}tE|| || z | |d |}|j|_#t t|_#| |_$| |k|_%|j|_||_&||_'||_(||_)gd|_*tW|S)Nr r0ddof_scale must be a non-negative number or Nonenaivez-when using weights, cov_type may not be naivegz)Singular matrix encountered in GEE updatermrHr dep_paramsFz,Iteration limit reached prior to convergencez"Unable to estimate GEE parameters.z del_paramsnum_assoc_updatesitrr;rrKncovrbc_covrr~res_kwdsresultss r<rOzGEE.fits   "ioa0DOO?? FHHH(DO,'-- < #G(;(;LMM M  //11KK:l33L&++--K   --- >> ' 'C 4466MFE~ I0222 6 !K  $ $[ 1 1 1 !3!344J  h ' . .{/?/?/A/A B B B  g & - -e 4 4 4  l + 2 2* , , , T!!&**do.F.F 'S<%7A$=$= 000"";///!Q&!    MH/ 1 1 1   M>, . . .4  dA < M23E G G G4 ~ % %__T**F ? &  ""A $ 7 7 T J J K" ,-?AAAt--a66GAty ,-?AAAt! 33Av>> 69M#01CEEE4##%%#'"&&,...T;u e&.e'/111 #/&t,,''$.!_+#3  !%%%!)))r>cbd\}}t|jD]}|j|\}} |j||z } t j|j|} |j|| dddfdzz} |j ||| | | f} | d}|t j | j |z }| d}|t j | j |z }t j |}t j||z|z dtj}||z||kz}||dz |zz}||||kzz }|||zz}|jdd|jddzxx|j|zz cc<||j|z|zz} tj||}nb#tjj$rKt j tj||}d}t-j|YnwxYw||z}||fS)Nr.rmrr z1Encountered singularity in regularized GEE update)rrrrr"rrhrrr r1r2r-absclipinfflatr(r+rrrrrr")r7rHpen_wt scad_paramepssnhmrrrr rexr8sn0hm0apclippedenr;rs r<_update_regularizedzGEE._update_regularized_sHBt~&& $ $A)!,IFAM!$v-E74;//7788Da44=!#33B?::!1dUBK99Dq'C "&s## #Bq'C "&s## #BB VF^^'*v-2Arv>> g f - zA~'' ff %% cBh !!"(1+/!"""dnr&99""" dnr!F** Y__R,,FFy$   VBINN2..33FEC M#       d!!###rzs F44AHHc||d}t|jD]}|j|\}}|j||z }t j|j|}|j ||dddfdzz}|j ||||f} t j |j | d} |t j| | z }|S)Nrrm)rrrrrr"rrhrrr r1r2r-r*) r7r>marrrr rr|r8ma0s r<_regularized_covmatzGEE._regularized_covmats   --- t~&& % %A)!,IFAM!$v-E74;//7788Da44=!#33B?::!1dUH66D&tAw''C "(3$$ $BB r>皙 @dh㈵>MbP?c | |_tj|jjd} || d} t t} ||jjd|_n|dkstd||_d} t|D]}| | |||\}}|d}tj |tn|| kr1tjtj|dz|krd } nl| |z } | d | || |dkr||zdkr|| | sd }tj |d| tj| |k<|| || }tj||}tj||j}t3d | }|} t7|| || d |}| |_t;|S)a\ Regularized estimation for GEE. Parameters ---------- pen_wt : float The penalty weight (a non-negative scalar). scad_param : float Non-negative scalar determining the shape of the Scad penalty. maxiter : int The maximum number of iterations. ddof_scale : int Value to subtract from `nobs` when calculating the denominator degrees of freedom for t-statistics, defaults to the number of columns in `exog`. update_assoc : int The dependence parameters are updated every `update_assoc` iterations of the mean structure parameter updates. ctol : float Convergence criterion, default is one order of magnitude smaller than proposed in section 3.1 of Wang et al. ztol : float Coefficients smaller than this value are treated as being zero, default is based on section 5 of Wang et al. eps : non-negative scalar Numerical constant, see section 3.2 of Wang et al. scale : float or string If a float, this value is used as the scale parameter. If "X2", the scale parameter is always estimated using Pearson's chi-square method (e.g. as in a quasi-Poisson analysis). If None, the default approach for the family is used to estimate the scale parameter. Returns ------- GEEResults instance. Note that not all methods of the results class make sense when the model has been fit with regularization. Notes ----- This implementation assumes that the link is canonical. References ---------- Wang L, Zhou J, Qu A. (2012). Penalized generalized estimating equations for high-dimensional longitudinal data analysis. Biometrics. 2012 Jun;68(2):353-60. doi: 10.1111/j.1541-0420.2011.01678.x. https://www.ncbi.nlm.nih.gov/pubmed/21955051 http://users.stat.umn.edu/~wangx346/research/GEE_selection.pdf r FNrrVz5Singular matrix encountered in regularized GEE updatermTrHz$GEE.fit_regularized did not convergerT)rZrE) regularizedr\)rr"zerosr8r(rrrrr&rrrrr rrr2rrersrr+rr-rr"rgr^ri)r7rwrxrdr update_assocrcztolryrr>r`r^miniterrmr;r{rrcovrpr8s r<fit_regularizedzGEE.fit_regularizeds{phtyq122    --- !$''   "ioa0DOO?? FHHH(DO >> 0 0C11)6:sDDJFB~M c#5666W}} ):):!;!;d!B!B  6 !K  ! ( ()9)9);); < < <  $ $[ 1 1 1axxS</144"";/// 8C M#   23 BF;''$./ ;'''  % %k 2 2ioob"%%ioob#%((c:::##%%$ S%&*h@@@& &&&r>c Pt|}|jjjd}tj|t j||z f}|j}|jj|_ddl }| |j }| || \} } | tjdt dS|\} } } |} | | dzz } | |d| z }tj| }| d|d|f}| |d|df}| d||df}|d|d|f}|d||df}|t j|jtj||z }|t j|jtj||z}|t j|jt jtj||tj||z }ddlm}t j|tj||}t|}d|||z }|||d|_|j|}|j|}||_||_ |j|_||fS) a Expand the parameter estimate `mean_params` and covariance matrix `bcov` to the coordinate system of the unconstrained model. Parameters ---------- mean_params : array_like A parameter vector estimate for the reduced model. bcov : array_like The covariance matrix of mean_params. Returns ------- mean_params : array_like The input parameter vector mean_params, expanded to the coordinate system of the full model bcov : array_like The input covariance matrix bcov, expanded to the coordinate system of the full model r rNrr/rmrr) r'rjr)r(r"r_rrrrrrrrrrr rr"r+rr2r-rrrrscore_test_resultsrIrLrEr8)r7r>rKred_pfull_p mean_params0 save_exog_lirsave_cached_meansrrrrrrrrrrrrr rr r rs r<rfzGEE._handle_constraints.K  $*1-u["(6E>*B*BBC | 8   MM$*;<<   ...++--5 = MI, . . .:5$##%%eqj uvv&y}}U##qw%'(uvvuvv~&qw'qw%'(qw'bfWY%'Y__Wg%F%FHHH RVGIIOOGW==?? ? RVGIF29??7G#D#D#%9??7G#D#DFFGG G  322222&!#F!C!CEEv;;488OX>>> 0?)1.:#<#<o22;?? ))$//# -O0022 D  r>c:|j|dS)z3 Update the association parameters N)r r;rGs r<rezGEE._update_assoc^s v&&&&&r>dydxcd}||j}|j}||||}d|vr||z}d|vr#||||dddfz}|ddlm}|||||||}|ddlm} | ||||||}|S)a1 For computing marginal effects, returns dF(XB) / dX where F(.) is the fitted mean. transform can be 'dydx', 'dyex', 'eydx', or 'eyex'. Not all of these make sense in the presence of discrete regressors, but checks are done in the results in get_margeff. Nr|eyr)_get_count_effects)_get_dummy_effects)r8rr,predict%statsmodels.discrete.discrete_marginsrr) r7rHr8 transform dummy_idx count_idxr+margeffrrs r<_derivative_exogzGEE._derivative_exoges <9D"3O&&tV_EE 9   tOG 9   t||FD11!!!T': :G   $ $ $ $ $ $(($ 9)-v77G   $ $ $ $ $ $(($ 9)-v77Gr>c|jj}g}d}t|jD]h}|j|\} } || ||j|| } |tj | j | |z z }itj |}tj |j } || z } tj |}tjdd|}t|D]B\}}| |dz| zdz z}||}tj| dz|dzz|z  ||<C|d|zz} ddlm}n#t&$r dd lm}YnwxYw|||d|dz }d |zd|jjdzz}d |zdtjtj ||zz}|||fS) aB Returns quasi-information criteria and quasi-likelihood values. Parameters ---------- params : array_like The GEE estimates of the regression parameters. scale : scalar Estimated scale parameter cov_params : array_like An estimate of the covariance matrix for the model parameters. Conventionally this is the robust covariance matrix. n_step : integer The number of points in the trapezoidal approximation to the quasi-likelihood function. Returns ------- ql : scalar The quasi-likelihood value qic : scalar A QIC that can be used to compare the mean and covariance structures of the model. qicu : scalar A simplified QIC that can be used to compare mean structures but not covariance structures Notes ----- The quasi-likelihood used here is obtained by numerically evaluating Wedderburn's integral representation of the quasi-likelihood function. This approach is valid for all families and links. Many other packages use analytical expressions for quasi-likelihoods that are valid in special cases where the link function is canonical. These analytical expressions may omit additive constants that only depend on the data. Therefore, the numerical values of our QL and QIC values will differ from the values reported by other packages. However only the differences between two QIC values calculated for different models using the same data are meaningful. Our QIC should produce the same QIC differences as other software. When using the QIC for models with unknown scale parameter, use a common estimate of the scale parameter for all models being compared. References ---------- .. [*] W. Pan (2001). Akaike's information criterion in generalized estimating equations. Biometrics (57) 1. rgwJr g@rmr) trapezoid)trapz)dx)rhrrrrr2r(rr"r2r-rremptylinspacerrscipy.integrater ImportErrorrr8r(trace)r7rHr cov_paramsn_steprmeansomegarrr4r'rduqvxvguvurqlqicuqics r<rzGEE.qicsh+&t~&& 2 2A+A.KFC LL ??4<?C88D RVDFD))E1 1EEu%%>$-00 X  Xf   [1f - -bMM 2 2DAqAER<#--ABVBEQUOb0111BqEE q5y ; 1 1 1 1 1 1 1 ; ; ; : : : : : : : : ;Yrbebem , , ,BwTY_Q///2gBHRVE:%>%>????3}sE EE) NNNrbNNNNTNrAr) rRrSNr rrTNrN)rrNrrrrSN)NrNN)r)"rMrNrO _gee_init_docrr_gee_family_doc _gee_examplerPrr= classmethodrrrr"r(r,rrrrMrQr _gee_fit_docrOrrrrfrerr __classcell__rs@r<r[r[s@ &0&     L>B9=:>*.`$`$`$`$`$`$H8<6:ooooo[ob 0 0 0N)N)N)`'''R46777r4448B+B+B+L)))V   Xl6:-.?AK*K*K*K*Z&&&P(?B67>Bs's's's'jL!L!L!\'''=C37""""HXXXXXXXXr>r[ceZdZdezZ d fd ZedZd!dZedZ dZ ed Z ed Z ed Z d"dZe Ze Ze Zeeddiz d#dZeeddizd$dZeeddiz d%dZd&dZd'dZ d(dZ d)dZdZeZxZS)*rgzHThis class summarizes the fit of a marginal regression model using GEE. rTFc pt|||||j|_|j|_|j|_|di} |j| t|drt|ds||_ |j } gd} | | vr'dd | z} t| |dkr|j } n|d kr|j} n |d kr|j} | |_dS|j |krtd dS) N)normalized_cov_paramsrr\rZr]rTrWrYzGEE: `cov_type` must be one of , rTrWrYz@cov_type in argument is different from already attached cov_type)rr=rrrhpop__dict__r;rrZlowerjoinr&rErDrLr])r7rrHrrrZr[rkwdsr\covariance_typeallowed_covariancesrrrs r<r=zGEEResults.__init__si  65      l HH["--  Y'''j)) >233 >$DM"m1133O"E"E"E &9998yy!4556 oo%8##oW$$n^++(&)D # # #}(( "=>>>)(r>c|jS)z) The response residuals. )resid_responser@s r<r zGEEResults.resids ""r>c|}gd}||vr'dd|z}t||dkr+tjtj|jS|dkr+tjtj|jS|dkrA|jtdtjtj|jSdS) a This is a convenience function that returns the standard errors for any covariance type. The value of `bse` is the standard errors for whichever covariance type is specified as an argument to `fit` (defaults to "robust"). Parameters ---------- cov_type : str One of "robust", "naive", or "bias_reduced". Determines the covariance used to compute standard errors. Defaults to "robust". rz&GEE: `covariance_type` must be one of rrTrWrYNz,GEE: `bias_reduced` covariance not available) rrr&r"rdiagrErDrL)r7rZrrrs r<standard_errorszGEEResults.standard_errorss #..**AAA "5 5 5;990112CS// ! h & &7274?3344 4  ' '7274>2233 3  . .!) BDDD7274#56677 7 / .r>c6||jSr)rrZr@s r<bsezGEEResults.bse<s##DM222r>cpt|jdsd}|dz }t||jjS)a Return the results of a score test for a linear constraint. Returns ------- Adictionary containing the p-value, the test statistic, and the degrees of freedom for the score test. Notes ----- See also GEE.compare_score_test for an alternative way to perform a score test. GEEResults.score_test is more general, in that it supports testing arbitrary linear equality constraints. However GEE.compare_score_test might be easier to use when comparing two explicit models. References ---------- Xu Guo and Wei Pan (2002). "Small sample performance of the score test in GEE". http://www.sph.umn.edu/faculty1/wp-content/uploads/2012/11/rr2002-013.pdf rz3score_test on results instance only available when z model was fit with constraints)rrr&r)r7rs r< score_testzGEEResults.score_test@s@0tz#788 "GC 4 4CS// !z,,r>cg}|jjD]4}|jj|}||j|5|S)z Returns the residuals, the endogeneous data minus the fitted values from the model. The residuals are returned as a list of arrays containing the residuals for each cluster. )rrrr2r r7sresidvrs r< resid_splitzGEEResults.resid_split_sL( * *A)!,B MM$*R. ) ) ) ) r>c|j}|jjD]<}|jj|}||xx||zcc<=|S)zC Returns the residuals centered within each group. )r rrrrmean)r7cresidrrs r<resid_centeredzGEEResults.resid_centeredlsc ""( , ,A)!,B 2JJJ&*//++ +JJJJ r>cg}|jjD]4}|jj|}||j|5|S)z Returns the residuals centered within each group. The residuals are returned as a list of arrays containing the centered residuals for each cluster. )rrrr2centered_residrs r<resid_centered_splitzGEEResults.resid_centered_splitwsN( 3 3A)!,B MM$-b1 2 2 2 2 r>Nrc|tjd||j}|j|j|||\}}}||fS)zh Returns the QIC and QICu information criteria. See GEE.qic for documentation. NzMQIC values obtained using scale=None are not appropriate for comparing models)r)rrrrrrHr)r7rrrrrs r<rzGEEResults.qicsr = M= > > > =JEz~~dk5&*oo&7&7-3&55 3Dyr>extra_params_docr\Tc4ddlm}|||||||}|S)Nr)plot_added_variable) resid_typeuse_glm_weights fit_kwargsax)$statsmodels.graphics.regressionplotsr)r7 focus_exogrrrrrfigs r<rzGEEResults.plot_added_variablesG MLLLLL!!$ -72A-7B@@@  r>c*ddlm}||||S)Nr)plot_partial_residuals)r)rr)r7rrrs r<rz!GEEResults.plot_partial_residualss/ POOOOO%%dJ2>>>>r>Q?c.ddlm}||||||S)Nr)plot_ceres_residuals) cond_meansr)rr)r7rfracrrrs r<rzGEEResults.plot_ceres_residualss@ NMMMMM##D*d/9bBBB Br>皙?c||j}n||}|j}tj}|d|dz z }||j||zz }|j||zz} nc ddddgfddgfd|jjjjgfd|jjjjgfd d |jgfg}d |jjD}d t|gfd t|jjgfdt|gfdt|gfddtj |zgfddt|j dzgfdd|jzgfdg}tj|j}tj|j} tj|j} tj|j} dd|zgfdd| zgfg} dd| zgfdd| zgfg} ||jjjdzdz}| |jj}| |jj}d d!lm}|}|||||||"|||||d#$||| | ||d"|S)%a| Summarize the GEE regression results Parameters ---------- yname : str, optional Default is `y` xname : list[str], optional Names for the exogenous variables, default is `var_#` for ## in the number of regressors. Must match the number of parameters in the model title : str, optional Title for the top table. If not None, then this replaces the default title alpha : float significance level for the confidence intervals cov_type : str The covariance type used to compute the standard errors; one of 'robust' (the usual robust sandwich-type covariance estimate), 'naive' (ignores dependence), and 'bias reduced' (the Mancl/DeRouen estimate). Returns ------- smry : Summary instance this holds the summary tables and text, which can be printed or converted to various output formats. See Also -------- statsmodels.iolib.summary.Summary : class to hold summary results )Dep. Variable:N)zModel:NMethod: Generalizedr\zEstimating EquationszFamily:zDependence structure:)zDate:NzCovariance type: c,g|]}t|SrQrr}s r<rxz&GEEResults.summary..s 2 2 2c!ff 2 2 2r>zNo. Observations:z No. clusters:zMin. cluster size:zMax. cluster size:zMean cluster size:z%.1fzNum. iterations:z%drHzScale:z%.3f)zTime:NzSkew:z%12.4fzCentered skew:z Kurtosis:zCentered kurtosis:N zRegression Resultsr)SummarygleftgrightynamexnametitleF)rrrr[)rrhrrMr rZrrr'minrr"rr^rrskewr kurtosisr exog_names endog_namesstatsmodels.iolib.summaryradd_table_2colsadd_table_params)r7rrrrtop_leftNY top_rightskew1kurt1skew2kurt2 diagn_left diagn_rightrsmrys r<summaryzGEEResults.summarysD-$0012!2!?A#(4=*;< 3 2dj1 2 2 2)CGG95%DJ,?(@(@'AB*SWWI6*SWWI6*Vbgbkk-A,BC(4+.t/?/I+J+J,K+LM$*!4 56$   4:&&tz** 4.//t233E!1 23'(U*:);<> $h&6%78,x%/?.@A  =J(1C7$%E =J)E =J*E 655555wyy T)#(#(  * * * d%u$)  8 8 8 T$/u#(  4 4 4 r>overallrcx|jjtjdtt ||||||fS)a{ Get marginal effects of the fitted model. Parameters ---------- at : str, optional Options are: - 'overall', The average of the marginal effects at each observation. - 'mean', The marginal effects at the mean of each regressor. - 'median', The marginal effects at the median of each regressor. - 'zero', The marginal effects at zero for each regressor. - 'all', The marginal effects at each observation. If `at` is 'all' only margeff will be available. Note that if `exog` is specified, then marginal effects for all variables not specified by `exog` are calculated using the `at` option. method : str, optional Options are: - 'dydx' - dy/dx - No transformation is made and marginal effects are returned. This is the default. - 'eyex' - estimate elasticities of variables in `exog` -- d(lny)/d(lnx) - 'dyex' - estimate semi-elasticity -- dy/d(lnx) - 'eydx' - estimate semi-elasticity -- d(lny)/dx Note that tranformations are done after each observation is calculated. Semi-elasticities for binary variables are computed using the midpoint method. 'dyex' and 'eyex' do not make sense for discrete variables. atexog : array_like, optional Optionally, you can provide the exogenous variables over which to get the marginal effects. This should be a dictionary with the key as the zero-indexed column number and the value of the dictionary. Default is None for all independent variables less the constant. dummy : bool, optional If False, treats binary variables (if present) as continuous. This is the default. Else if True, treats binary variables as changing from 0 to 1. Note that any variable that is either 0 or 1 is treated as binary. Each binary variable is treated separately for now. count : bool, optional If False, treats count variables (if present) as continuous. This is the default. Else if True, the marginal effect is the change in probabilities when each observation is increased by one. Returns ------- effects : ndarray the marginal effect corresponding to the input options Notes ----- When using after Poisson, returns the expected number of events per period, assuming that the model is loglinear. Nz#marginal effects ignore constraints)rrjrrr GEEMargins)r7atmethodatexogdummycounts r< get_margeffzGEEResults.get_margeffFsGz : , M?& ( ( ($VVUE BCCCr> 2cddlm}|j|j}|j|jj}ggct ||D]\}}tj|j dd} || d|| dz|j dzz } |tj || dddf|| dddfz dz d}  | tjtj|||\} }n|} tjdk} tj| } | ztj|r#tjdt+|}tj|tj}tjfd|D}tj||k} || }tjfd|D}tjfd|D}|||d d d |d |d| S)a6 Create a plot of the pairwise products of within-group residuals against the corresponding time differences. This plot can be used to assess the possible form of an isotropic covariance structure. Parameters ---------- ax : AxesSubplot An axes on which to draw the graph. If None, new figure and axes objects are created xpoints : scalar or array_like If scalar, the number of points equally spaced points on the time difference axis used to define bins for calculating local means. If an array, the specific points that define the bins. min_n : int The minimum sample size in a bin for the mean residual product to be included on the plot. rutilsr rmNc@g|]}tj|kSrQ)r"r)rtrudgs r<rxz8GEEResults.plot_isotropic_dependence..s%888q26"'??888r>cLg|] }tj|k!SrQr"r)rtrur7xres r<rxz8GEEResults.plot_isotropic_dependence..+===A"'#bAg,//===r>cLg|] }tj|k!SrQr9)rtrur7xdts r<rxz8GEEResults.plot_isotropic_dependence..r;r>-oranger)colorlwzTime differencezProduct of scaled residuals)statsmodels.graphicsr5rrr rVzipr" tril_indicesr(rr2rrr create_mpl_ax get_figure flatnonzerorrrrdigitizerrsplot set_xlabel set_ylabel)r7rxpointsmin_ngutilsr rVretirdistsrrv0dguhistdgydgxr7r=r:s @@@r<plot_isotropic_dependencez$GEEResults.plot_isotropic_dependences. 988888 '' 33z&&tz77rS%&&  FB!a00BBqER1Y&q8B JJrNNNGbAlR1qqq\9a?DDQGGHHE JJu    nS!!nS!! :**2..GC--//C^C1H % % WSW   r  ;w   8k!SXXw77G [g & &immz8888C88899 ^DEM * *"gj========>>j========>> S#X!444 '((( 3444 r>c|j}ddl}||jj}|j}d|_g}g} t jdd|D]} | |zd| z |zz} || |||_| |j_| |j |j |j |j |j} | | ||_| S)a Refits the GEE model using a sequence of values for the dependence parameters. Parameters ---------- dep_params_first : array_like The first dep_params in the sequence dep_params_last : array_like The last dep_params in the sequence num_steps : int The number of dep_params in the sequence Returns ------- results : array_like The GEEResults objects resulting from the fits. rNFr )rjrcrarbrZ)rrrr rir"rr2rXrOrHrcrarbrZ) r7dep_params_firstdep_params_last num_stepsrrr rirXrqrdpr8s r<sensitivity_paramszGEEResults.sensitivity_paramss*  ]]4:#899 %   Q9-- ! !A_$A1A'AAB   b ! ! !#}}Z88E *,E  '99$+"&)*.*;.2.C&*m 55D NN4 %r>)rTFF)rT)Nr)NTNNr)rNN)rNN)NNNrr(rNFF)Nr1r2) rMrNrO_gee_results_docrPr=rr rrrrrrr split_residrsplit_centered_residrrrrrrrrr'r0rWr]params_sensitivityrrs@r<rgrgse ) * >C'>'>'>'>'>'>R##^# 8888D33^3--->  ^ ^  ^ .K#N/ X&*rgcLeZdZddiZejejjeZdS)rirrowsN) rMrNrO_attrswrap union_dictslmRegressionResultsWrapper _wrap_attrsrQr>r<riri s;&F#$"2#>#J#)++KKKr>riceZdZdeejeeedzzZ dfd Z dZ dZ e e dfd ZxZS) OrdinalGEEz9 Ordinal Response Marginal Regression Model using GEE r]Nrbc B|tj}n)t|tjstd|t j}||||||\}}}}}t||||||||| | dS)Nz&ordinal GEE must use a Binomial family) rrrr&rOrdinalIndependence setup_ordinalrr= r7rTr8rUrVrhr rgrWrfrjrrs r<r=zOrdinalGEE.__init__ s >&((FFfh&788 K !IJJJ  $8::J,0,>,> 4v-/-/)tVT6 fdW: 7 7 7 7 7r>c||_||_||_|||_n(d|_t jt|}|||_n*d|_t jt|df}t j |}t j |}t j |}t j |}t j |}t j ||_ |j dd}t|}|t|z}t j||j dft j } t j|t j } t j||ft j } t j||j} t j||j dft j } t j|t j }d}t|||||}|D]i\}}}}}t!|D]Q\}}|| |ddf<t#t j||k| |<d| ||f<|| |<|| |<|||<|dz }Rjt j| | fd} d|D}t)|jt*jur ||jjn;|dt3d|j ddzDt+j| | } t)|jt*jur t+j| |jj } | | | | |fS) z Restructure ordinal data as binary indicators so that they can be analyzed using Generalized Estimating Equations. Nr rrz)axiscg|]}d|zS)z I(y>%.1f)rQ)rtrs r<rxz,OrdinalGEE.setup_ordinal..g s666a+/666r>cg|]}d|zSzx%drQrtrus r<rxz,OrdinalGEE.setup_ordinal..k sJJJ519JJJr>rname)r endog_orig exog_orig groups_orig offset_origr"rr' time_origrsr endog_valuesr(r|rqrCrror$rrr DataFramerrrrrz)r7rTr8rUrVrW endog_cutsncutnrowsexog_out endog_out intercepts groups_outtime_out offset_outjrowzipperexog_row endog_value group_value time_value offset_value thresh_ixthreshxnamess r<rozOrdinalGEE.setup_ordinal+ s  **,,!;;==  %{{}}D  #D Xc%jj))F  !YY[[DNN!DN8SZZO,,Dz$ 5!!F##z$F##Ie,,&qt, :s5zz!8UDJqM2"$*...HU"*555 Xudm2:>>> Xe6<888 8UDJqM2"$*...Xe2:666 T5&$77$   X{K &/z%:%:  ! 6$,qqq!"%bjv1E&F&F"G"G $./ 4?+#. 4 !+#/ 4   >:x"8qAAA76:666   2< / / MM$.0 1 1 1 1 MMJJeAtz!}q7H.I.IJJJ K K K<&999  BI - - )$/2FGGGI(J*DDr>c t|dd}t|j|j|j|jt j|j|}| }|j SNrX)rVrhrWrX rr[rTr8rUrVrrrWrOrHr7rXrrPs r<rQzOrdinalGEE._starting_paramst a4T22DJ 4;8+<+>+>;;;;}r>rRrSr rrTc t|||||| j fd jD}t | j  jz j||}t|S)Nrc2i|]}|t|SrQrrtrur8s r< z"OrdinalGEE.fit.. %===AAwtQ''===r>rZr\) rrO_resultsrhOrdinalGEEResultsrHrrOrdinalGEEResultsWrapper) r7rdrcrjrarbrZrpord_rsltr8rs @r<rOzOrdinalGEE.fit| s ww{{7D,')9$,..}=======$T4;%)__%6%6%C%)Z.6/7 999(111r>NNNrbNNNrRrSNr rrT)rMrNrOrrr_gee_ordinal_family_doc_gee_ordinal_example_gee_nointerceptrPr=rorQrrrOrrs@r<rlrl s E)@'>$8"244 4 4 ?C9=+/777777(GEGEGERXl6:-.2222222222r>rlc eZdZdezZddZdS)rz_This class summarizes the fit of a marginal regression modelfor an ordinal response using GEE. Nc <ddlm}|||\}}n|}|ig}|jjd}dt|jjD}|D][}| D]"}||jjvrtd|z#g} |D]} tj |j } d| | <t|jjD]%\} } | |vr | |vr || | | <|| | | <&ddtjtj| |j  zz }| || dd| dtj| } tj|  }||jj|d]|d |d |dd|S) a^ Plot the fitted probabilities of endog in an ordinal model, for specified values of the predictors. Parameters ---------- ax : AxesSubplot An axes on which to draw the graph. If None, new figure and axes objects are created exog_values : array_like A list of dictionaries, with each dictionary mapping variable names to values at which the variable is held fixed. The values P(endog=y | exog) are plotted for all possible values of y, at the given exog value. Variables not included in a dictionary are held fixed at the mean value. Example: -------- We have a model with covariates 'age' and 'sex', and wish to plot the probabilities P(endog=y | exog) for males (sex=0) and for females (sex=1), as separate paths on the plot. Since 'age' is not included below in the map, it is held fixed at its mean value. >>> ev = [{"sex": 1}, {"sex": 0}] >>> rslt.distribution_plot(exog_values=ev) rr4NcBg|]\}}|d|S)zI() startswith)rtrrs r<rxz7OrdinalGEEResults.plot_distribution.. s<((($!QLL&&(A(((r>!%s is not a variable in the modelrr o-Response value Probability)rBr5rErFrr8rrrkeysr&r" zeros_likerHexpr2r2insertrsdiffrIrrJrKset_ylim)r7r exog_valuesrNr exog_meansix_iceptevruprjxprvnpprds r<plot_distributionz#OrdinalGEEResults.plot_distribution sC< 988888 :**2..GC--//C  &KZ_))!,, (()DJ,A"B"B((( 8 8BWWYY * *DJ111$%H'(&)***2 B  ]4;//1&tz'<==..EArH}} Rxx "21!+1 1RVRVB %<%<$<===> ! IIaOOO IIaLLLBB72;;,C GGDJ+S$ 7 7 7 7 &''' m$$$ Aq r>r/rMrNrOr_rPrrQr>r<rr s@ /   TTTTTTr>rc |j}|j}tj|d\}}}|j}tj|\}} |tj|tj|j|z }tj|d\} } } | dd| dkf} tjtj|dk} tj|dd| ftj|dd| fj||| dfz } tj tj|tj|| z }|dkrdStj|dd| ftj|dd| fj| || dfz }| |fS)ac Return transformation matrices for design matrices. Parameters ---------- par : instance The parent model sub : instance The sub-model Returns ------- qm : array_like Matrix mapping the design matrix of the parent to the design matrix for the sub-model. qc : array_like Matrix mapping the design matrix of the parent to the orthogonal complement of the columnspace of the submodel in the columnspace of the parent. Notes ----- Returns None, None if the provided submodel is not actually a submodel. rNg-q=g:0yE>r/) r8r"r+r,r-qrr2rGrsr)parsubx1x2rsvtrarx2csbrrers r<rr s4 B By}}R##HAq" A 9<<  DAq BF1bfQS!nn % %%Aq!$$JCQ aaaem C q E) * *B !!!R%"&111b5R001RX;> ? ?B rvb26"b>>)**++A4xxz !!!R%"&111b5S11Ab$hK? @ @B r6Mr>ceZdZdS)rNrMrNrOrQr>r<rr& Dr>rceZdZdeejeeedzzZ dfd Z dZ dZ dZ dd Zee dfd ZxZS) NominalGEEz: Nominal Response Marginal Regression Model using GEE. r]Nrbc ||||||\}}}}}|t|jdz}|tj}t ||||||||| | dSNr ) setup_nominalrrrNominalIndependencerr=rps r<r=zNominalGEE.__init__4 s-1,>,> 4v-/-/)tVT6 >!$)a-00F  $8::J  4vz7 Hj * * * * *r>c t|dd}t|j|j|j|jt j|j|}| }|j Srrrs r<rQzNominalGEE._starting_paramsE rr>c||_||_||_|||_n(d|_t jt|}|||_n*d|_t jt|df}t j |}t j |}t j |}t j |}t j |}t j ||_ |j dd}t|}||_ t||j dz}t||j dz} t j|| ft j} t j|t j} t j|t j} t j||j dft j} t j|t j}d}t|||||}|D]\}}}}}t!|D]}\}}t jt|t j}d||<t j||| |ddf<t%||k| |<|| |<|| |<|||<|dz }~t'|jt(jr |jj}n(dt/d|j ddzD}g}|D]#|fd|D$t)j| |} t)j| |} t'|jt(jr t)j| |jj } | | | | |fS) z Restructure nominal data as binary indicators so that they can be analyzed using Generalized Estimating Equations. Nr rrrrzcg|]}d|zSrvrQrws r<rxz,NominalGEE.setup_nominal.. sHHHqHHHr>c"g|] }|ddd S)[z.1f]rQ)rtrtrs r<rxz,NominalGEE.setup_nominal.. s-???a++"++++???r>rxry)rr{r|r}r~r"rr'rrsrrrr(r|rCrkronrorrrrrrrrz)r7rTr8rUrVrWrrrncolsrrrrrrrrrrrrrrr xnames_inrrs @r<rzNominalGEE.setup_nominalM s  **,,!;;==  %{{}}D  #D Xc%jj))F  !YY[[DNN!DN8SZZO,,Dz$ 5!!F##z$F##Ie,,&qt, : J$*Q-/J$*Q-/8UEN"*===HU"*555 Xe2:666 8UDJqM2"$*...Xe2:666 T5&$77$   X{K &/z%:%:  ! 6HS__BJ??? ) $&GAx$8$8qqq!#&{f'<#=#= $#. 4 !+#/ 4    dnbl 3 3 I.IIHHE!TZ]Q5F,G,GHHHI A AB MM????Y??? @ @ @ @<&999<&999 dory 1 1 H )$/2FGGGI(J*DDr>ctj|}tj|t||jz|jf}d|dz}tj|tj|jtj}||z }|dddf|z}|dddf|z}||dddf|z|dddfz z}|S)a Derivative of the expected endog with respect to the parameters. Parameters ---------- exog : array_like The exogeneous data at which the derivative is computed, number of rows must be a multiple of `ncut`. lin_pred : array_like The values of the linear predictor, length must be multiple of `ncut`. Returns ------- The derivative of the expected endog with respect to the parameters. r rzN r"rreshaper'rrronesr|) r7r8r%rexpval_mdenommprobr'ddenoms r<r(zNominalGEE.mean_deriv s&!!:fs6{{di'?'+y'233HLLOO#rwty CCCDDQQQW~$44' aaag'%4.88 r>c |tjdttj||}tj|}tj|t||jz|jf}d| dz}tj |tj |jtj }tj tj |jd|}g} t|jD]{} tj|jtj } d| | <| tj | tj t||jz|tj| } tj tj |jd|jzdf| } || z} |dddf| z|dddfz } tj |tj |jdf}tj |tj d|jf}| |dddf| |zz|dddfdzz z} | S)a Derivative of the expected endog with respect to exog for the multinomial model, used in analyzing marginal effects. Parameters ---------- exog : array_like The exogeneous data at which the derivative is computed, number of rows must be a multiple of `ncut`. lpr : array_like The linear predictor values, length must be multiple of `ncut`. Returns ------- The value of the derivative of the expected endog with respect to exog. Notes ----- offset_exposure must be set at None for the multinomial family. Nz2Offset/exposure ignored for the multinomial familyr rzrrm)rrrr"r2rrr'rrrrr|r*r(rrr2r)r7r8rHr+r4rrrbmat0qmatreer3r' expval_mbs r<r,zNominalGEE.mean_deriv_exog s-0  & MN& ( ( (fT6"":fs6{{di'?'+y'233HLLOO#rwty CCCDDA//88ty!! H HA$)2:666BBqE KKBGCKK49,D$E$EFF G G G Gx~~wrw 1  :A>??FFt|aaag%aaag6GHbgty!n&=&=>> GIrw49~'>'>??  qqq$w4)#34uQQQW~7JJJ r>rRrSr rrTcT t|||||| tjdtdS j fd jD}t| j j z j ||}t|S)NrzGEE updates did not convergec2i|]}|t|SrQrrs r<rz"NominalGEE.fit.. rr>r) rrOrrr rrhNominalGEEResultsrHrrNominalGEEResultsWrapper) r7rdrcrjrarbrZrpnom_rsltr8rs @r<rOzNominalGEE.fit s ww{{7D,')9$,.. < M8, . . .4}=======$T4;%)__%6%6%C%)Z.6/7 999(111r>rrr)rMrNrOrrr_gee_nominal_family_doc_gee_nominal_examplerrPr=rQrr(r,rrrOrrs@r<rr+ s F)@'>$8"244 4 4 ?C9=+/******"IEIEIEV)))V8888tXl6:-.2222222222r>rc eZdZdezZddZdS)rz^This class summarizes the fit of a marginal regression modelfor a nominal response using GEE. Ncddlm}|||\}}n|}|ig}|jjjj}|jjj}t|jj j d|z }|jj dd|}|jj d|} d| D} tj|j|t#|j|zf} |D]} |} | D]8}|| vrt)d|z| |} | || | <9tj| | }||}tj|d|z f}||jj|d|d|d ||jj||jj|dd|S) a^ Plot the fitted probabilities of endog in an nominal model, for specified values of the predictors. Parameters ---------- ax : AxesSubplot An axes on which to draw the graph. If None, new figure and axes objects are created exog_values : array_like A list of dictionaries, with each dictionary mapping variable names to values at which the variable is held fixed. The values P(endog=y | exog) are plotted for all possible values of y, at the given exog value. Variables not included in a dictionary are held fixed at the mean value. Example: -------- We have a model with covariates 'age' and 'sex', and wish to plot the probabilities P(endog=y | exog) for males (sex=0) and for females (sex=1), as separate paths on the plot. Since 'age' is not included below in the map, it is held fixed at its mean value. >>> ex = [{"sex": 1}, {"sex": 0}] >>> rslt.distribution_plot(exog_values=ex) rr4Nr cDg|]}|ddS)rr)splitrs r<rxz7NominalGEEResults.plot_distribution..P s&:::!aggcll1o:::r>rrrr) rBr5rErFrrhrr=rror8r(rrr"rrHr'rrr&rpr2rrrIrrJrK set_xticksset_xticklabelsr)r7rrrNrrrrurrrHrr8rr4rs r<rz#NominalGEEResults.plot_distribution sG< 988888 :**2..GC--//C  &Kz %-z %  %a(4/ 0 0Z_))!,,QqS1 Z*1Q3/ ::z::: DK!3t{#3#3t#;<>> 7 7B??$$DWWYY ! !J&&$%H'(&)*** %%a((a5R&&&CcBr1rvvxx<'(B GGDJ+R 6 6 6 6 &''' m$$$ dj-... 4:2333 Aq r>r/rrQr>r<rr s@ .   MMMMMMr>rceZdZdS)rNrrQr>r<rrp rr>rceZdZdZdZdZdS)_MultinomialLogitac The multinomial logit transform, only for use with GEE. Notes ----- The data are assumed coded as binary indicators, where each observed multinomial value y is coded as I(y == S[0]), ..., I(y == S[-1]), where S is the set of possible response labels, excluding the largest one. Thererefore functions in this class should only be called using vector argument whose length is a multiple of |S| = ncut, which is an argument to be provided when initializing the class. call and derivative use a private method _clean to trim p by 1e-10 so that p is in (0, 1) c||_dSr)r)r7rs r<r=z_MultinomialLogit.__init__ s  r>c6tj|}dtj|t||jz|jfdz}tj|tj|jtj}||z }|S)a Inverse of the multinomial logit transform, which gives the expected values of the data as a function of the linear predictors. Parameters ---------- lpr : array_like (length must be divisible by `ncut`) The linear predictors Returns ------- prob : ndarray Probabilities, or expected values r rzr)r7r4rrprobs r<r=z_MultinomialLogit.inverse s"BJvF ty(@(, (34447CFF;rwty CCCDD~ r>N)rMrNrOrPr=r=rQr>r<rru s<"r>rc:eZdZdZegZejZegZ ddZ dZ dS)rzw Pseudo-link function for fitting nominal multinomial models with GEE. Not for use outside the GEE class. Tc>||_||dS)z Parameters ---------- nlevels : int The number of distinct categories for the multinomial distribution. N) _check_linkr)r7nlevels check_links r<r=z_Multinomial.__init__ s$&      r>cL|dz |_t|j|_dSr)rrr)r7rs r<rz_Multinomial.initialize s"aK %di00 r>N)T) rMrNrOrPrlinksvarfuncsbinaryrrr=rrQr>r<rr sZ  !EH#&J ! ! ! !11111r>rcreZdZdZifdZdZedZddZedZ ddZ dd Z ddZ d S)r*a Estimated marginal effects for a regression model fit with GEE. Parameters ---------- results : GEEResults instance The results instance of a fitted discrete choice model args : tuple Args are passed to `get_margeff`. This is the same as results.get_margeff. See there for more information. kwargs : dict Keyword args are passed to `get_margeff`. This is the same as results.get_margeff. See there for more information. c<i|_||_|j|i|dSr)_cacherqr0)r7rqrrs r<r=zGEEMargins.__init__ s.  $)&)))))r>ci|_dSr)rr@s r<_resetzGEEMargins._reset s  r>cHt|j|j|jz Sr)rmargeff_optionsr margeff_ser@s r<tvalueszGEEMargins.tvalues s"-...|do--r>rc t|jddlm}t|jddddddg}|jjjddk|jjj }fd t|D}tj |j |j|j|j||f}|||| S) aa Returns a DataFrame summarizing the marginal effects. Parameters ---------- alpha : float Number between 0 and 1. The confidence intervals have the probability 1-alpha. Returns ------- frame : DataFrames A DataFrame summarizing the marginal effects. r)rr,z Std. Err.zzPr(>|z|)zConf. Int. LowzCont. Int. Hi.c*g|]\}}| |SrQrQ)rtrrzinds r<rxz,GEEMargins.summary_frame.. s&IIIga#a&ITIIIr>)rrp)rrpandasrrrqrr8varrrr" column_stackrrrpvaluesr)r7rrnamesr var_namestablers @r< summary_framezGEEMargins.summary_frame s -...$$$$$$!$"6x"@Ac:!#35l %))!,,1\'2 IIII:)>)>III t !%t}}U/C/C!EFFyY????r>ct|jtjt j|jdzS)Nrm)rrrrsfr"rsrr@s r<rzGEEMargins.pvalues s8-...z}}RVDL1122Q66r>ct|j|j}tjd|dz z }|j||zz }|j||zz}tjt||S)a Returns the confidence intervals of the marginal effects Parameters ---------- alpha : float Number between 0 and 1. The confidence intervals have the probability 1-alpha. Returns ------- conf_int : ndarray An array with lower, upper confidence intervals for the marginal effects. r rm) rrrrrrrr"rsr)r7rme_serrrs r<rzGEEMargins.conf_int sq -... JNN1uqy= ) ) q5y( q5y(z$ue,,---r>c t|j|j}|j}|jjdz}|jd}d|jgfd|gfd|jdgfg}ddlm}m }m } |j d d } |} |j j } | | | tt!|d d } | d kr!||jd d \}}n |j}|g}| ||g|| |g}||}|j}|j}|j}|j}| d krt1| D]}||d d |f|d d |f|d d |f|d d |f|d d d d |ff}||||| |dd }|||_dt4|ddddd|dzz zg}|d|||| |d }nK||||||f}|||| |dd }dt4|ddddd|dzz zg}|d|| j|| S)aF Returns a summary table for marginal effects Parameters ---------- alpha : float Number between 0 and 1. The confidence intervals have the probability 1-alpha. Returns ------- Summary : SummaryTable A SummaryTable instance z Marginal Effectsr,r r zAt:r+r)rsummary_params table_extendNJr T)rrF)rrrr[ skip_headerr\zstd errrzP>|z|z[%3.1f%% Conf. Int.]r) keep_headers)rrrqrrrMrrrr&r'rr const_idxrror_get_endog_namerrrrrrrrrinsert_header_rowr2tables)r7rrqrrr,rrr&r'rr&r+r(r yname_listrrrrrreqrestuptbleheaders r<r'zGEEMargins.summary sS -..., (+>>%h/%(9':;)T1$789= = = = = = = = = = =%aaa( wyyJ(  NN9 % % % sA&& ' ' q55 ' 7 78I8<$!8!H!H E::%EJ T"#( %  I I I =='',_ ,, q55Ahh # #!7111b5>:aaae3D!!!!R%.'!!!R%.(111aaa8:LN%~fJrN,6e,126888(^ .v6 3!0C%#+4EFH&&q&111 T"""" LT:::EEw GWhOF"N6j).eOOOE*62Is5us{9JKMF  # #Av . . . 5!!! r>r(rNFc ~||}|}t||t|||_|j}|j}|j}|j } | ddk} |j j } |r$t||t| | \} }nd} |r$t||t| | \} }nd} t!| ||| } ||| || | }t%||}|dkr|dd| f|_dSt)||| |||j| | |d \}}|| dd| f|_|| |_|| |_dS)N)r,r+rrr )rrrrrrqrrHr8rrrr+rrrrrrrrr margeff_covr)r7r+r,r-r.r/rqrrHr8 effects_idxr+rreffectsr5rs r<r0zGEEMargins.get_margefff s  XXZZB'''#6b999, z  hhqkkQ& J(   V , , ,/i@@ IuuI   V , , ,/i@@ IuuI!r6;??((v)2I??gr** ;;"111k>2DLLL':vtW%7%7%9%92& 9'' #K +;7;GD (5DO";/DLLLr>)rr^) rMrNrOrPr=rrrr rrr'r0rQr>r<r*r* s  .0**** ..^.@@@@677^7.....PPPPd?C',101010101010r>r*)VrPstatsmodels.compat.pythonrstatsmodels.compat.pandasrnumpyr"scipyrrrr collectionsrstatsmodels.tools.decoratorsrstatsmodels.base.modelrr#statsmodels.regression.linear_model regression linear_modelrhstatsmodels.base.wrapperwrapperrfstatsmodels.genmodr+statsmodels.genmod.generalized_linear_modelr r r r$statsmodels.genmod.families.varfuncsgenmodr !statsmodels.genmod.families.linksr statsmodels.tools.sm_exceptionsr rrrr)statsmodels.graphics._regressionplots_docrrrrrrrrrrrrrrrrrrrrr_rrrrYr[rgriripopulate_wrapperrlrrrrrrrrrr*rQr>r<rLsj0+*****...... ######777777%%%%%%%%%000000000'''''''''''''''GGGGGGGG888888777777777777222222;;;;;;;;;;;;(((((((((((((((((((((( g<g<g<g<g<g<g<g