[scikit-learn] biased predictions in logistic regression

Rachel Melamed melamed at uchicago.edu
Sun Dec 18 17:04:08 EST 2016 

Josef!  Thank you!  You are a gift to the python statistics world.  I don’t know why I did not think of that. That is the answer.
Rachel

On Dec 18, 2016, at 10:09 AM, josef.pktd at gmail.com<mailto:josef.pktd at gmail.com> wrote:



On Sat, Dec 17, 2016 at 10:25 PM, Rachel Melamed <melamed at uchicago.edu<mailto:melamed at uchicago.edu>> wrote:
Hi Sean, Sebastian, Alexey (and Josef),
I’m not sure I fully understand what normalizing a dummy should consist of, so please let me know if I am interpreting your suggestion right.  I believe I can’t use the StandardScaler since I  am using grouped data. As I mentioned, I’m using a matrix “success_fail” containing the number of observations of each outcome per row of the  design matrix.  I tried to implement your suggestions by making my design matrix per each regression as previously, using patsy, but then normalizing:

        darray = np.array(design)
weights = np.tile(success_fail.sum(axis=1)/success_fail.sum().sum(),
(darray.shape[1],1)).transpose()

        dmean = (darray*weights).sum(axis=0)
        dvar = (((darray - dmean)**2)*weights).sum(axis=0)**.5
        dvar[dvar==0] = 1
        design_norm = (darray - dmean) / dvar
design_norm[:,0] = 1 ## intercept stays at 1

Then I use the normalized version of the design matrix as input to the regression. It seems like the bias is still there, though the results are a bit different (see plot)
Is this what you were suggesting? I’m also not sure I understand why the error would be higher on the training data for the less-regularized setting.
Thanks again
Rachel


Doing a partial check on the math:
AFAICS: The estimating equation or gradient condition for the intercept is unchanged even when the other parameters are penalized. This means that in the training sample the fraction of success or failures should coincide with the predicted probabilities of success or failures. The intercept should compensate for the penalization of the other parameters and for any transformation or standardization of the other explanatory variables.

AFAICS from your code snippets. You are using the intercept as part of X and fit_intercept=False.
This means your intercept is penalized. If you use the built-in fit_intercept=True, then the intercept is not penalized and the bias should go away.

Josef


On Dec 15, 2016, at 5:05 PM, Sean Violante <sean.violante at gmail.com<mailto:sean.violante at gmail.com>> wrote:

Sorry just saw you are not using the liblinear solver,  agree with Sebastian,  you should subtract mean not median

On 15 Dec 2016 11:02 pm, "Sean Violante" <sean.violante at gmail.com<mailto:sean.violante at gmail.com>> wrote:
The problem is the (stupid!)  liblinear solver that also penalises the intercept (in regularisation) . Use a different solver or change  the intercept_scaling parameter

On 15 Dec 2016 10:44 pm, "Sebastian Raschka" <se.raschka at gmail.com<mailto:se.raschka at gmail.com>> wrote:
Subtracting the median wouldn’t result in normalizing the usual sense, since subtracting a constant just shifts the values by a constant. Instead, for logistic regression & most optimizers, I would recommend subtracting the mean to center the features at mean zero and divide by the standard deviation to get “z” scores (e.g., this can be done by the StandardScaler()).

Best,
Sebastian

> On Dec 15, 2016, at 4:02 PM, Rachel Melamed <melamed at uchicago.edu<mailto:melamed at uchicago.edu>> wrote:
>
> I just tried it and it did not appear to change the results at all?
> I ran it as follows:
> 1) Normalize dummy variables (by subtracting median) to make a matrix of about 10000 x 5
>
> 2) For each of the 1000 output variables:
> a. Each output variable uses the same dummy variables, but not all settings of covariates are observed for all output variables. So I create the design matrix using patsy per output variable to include pairwise interactions.  Then, I have an around 10000 x 350 design matrix , and a matrix I call “success_fail” that has for each setting the number of success and number of fail, so it is of size 10000 x 2
>
> b. Run regression using:
>
> skdesign = np.vstack((design,design))
>
> sklabel = np.hstack((np.ones(success_fail.shape[0]),
> np.zeros(success_fail.shape[0])))
>
> skweight = np.hstack((success_fail['success'], success_fail['fail']))
>
>         logregN = linear_model.LogisticRegression(C=1,
>                                     solver= 'lbfgs',fit_intercept=False)
>         logregN.fit(skdesign, sklabel, sample_weight=skweight)
>
>
>> On Dec 15, 2016, at 2:16 PM, Alexey Dral <aadral at gmail.com<mailto:aadral at gmail.com>> wrote:
>>
>> Could you try to normalize dataset after feature dummy encoding and see if it is reproducible behavior?
>>
>> 2016-12-15 22:03 GMT+03:00 Rachel Melamed <melamed at uchicago.edu<mailto:melamed at uchicago.edu>>:
>> Thanks for the reply.  The covariates (“X") are all dummy/categorical variables.  So I guess no, nothing is normalized.
>>
>>> On Dec 15, 2016, at 1:54 PM, Alexey Dral <aadral at gmail.com<mailto:aadral at gmail.com>> wrote:
>>>
>>> Hi Rachel,
>>>
>>> Do you have your data normalized?
>>>
>>> 2016-12-15 20:21 GMT+03:00 Rachel Melamed <melamed at uchicago.edu<mailto:melamed at uchicago.edu>>:
>>> Hi all,
>>> Does anyone have any suggestions for this problem:
>>> http://stackoverflow.com/questions/41125342/sklearn-logistic-regression-gives-biased-results
>>>
>>> I am running around 1000 similar logistic regressions, with the same covariates but slightly different data and response variables. All of my response variables have a sparse successes (p(success) < .05 usually).
>>>
>>> I noticed that with the regularized regression, the results are consistently biased to predict more "successes" than is observed in the training data. When I relax the regularization, this bias goes away. The bias observed is unacceptable for my use case, but the more-regularized model does seem a bit better.
>>>
>>> Below, I plot the results for the 1000 different regressions for 2 different values of C:
>>>
>>> I looked at the parameter estimates for one of these regressions: below each point is one parameter. It seems like the intercept (the point on the bottom left) is too high for the C=1 model.
>>>
>>>
>>>
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>>>
>>>
>>> --
>>> Yours sincerely,
>>> Alexey A. Dral
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>> --
>> Yours sincerely,
>> Alexey A. Dral
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