I am studying the correlation between several human traits. One way is to use a chi-square test, but this is unable to include covariates. I am also using logistical regression to do this, and this makes it possible to include age and race as covariates.
However, I noticed that some tests support stratified data for a chi-square-like test.
Therefore, I am wondering what are the differences between including covariates in logistic regression and a stratified chi-square test?
Related
I use ANOVA, including the Levene test, unidimensional significance tests, descriptive statistics and the Tukey test. I have some doubts about what happens when the assumption of homogeneity of variance is not met in the Levene test. In many of the available materials I found the information:
"Basic assumptions of ANOVA tests:
Independence of random variables in the populations (groups) under consideration.
Measurability of the analysed variables.
Normality of the distribution of the variables in each population (group).
Homogeneity of the variance in all populations (groups).
If one of the first three assumptions is not met in the analysis of variance, the non-parametric Kruskal-Wallis test should be used. If the assumption of homogeneity of variances is not met, the Welch test should be used to assess the means."
If we had heterogeneous variances and a non-normal distribution then we would apply the Kruskal-Wallis test, right? On the other hand, what if we have both heterogeneous variances and a normal distribution, do we use the Welch test? If we do the Welch test, how should it be interpreted and what tests are subsequently recommended to see statistically significant differences between groups?
I would be very grateful for an answer
Multiple Linear Regression: a Significant ANOVA but NO significant coefficient predictors?
I have ran a multiple regression on 2 IVs to predict a dependant, all assumptions have been met, the ANOVA has a significant result but the coefficient table suggests that none of the predictors are significant?
what does this mean and how am I able to interpret the result of this?
(USED SPSS)
This almost certainly means the two predictors are substantially correlated with each other. The REGRESSION procedure in SPSS Statistics has a variety of collinearity diagnostics to aid in detection of more complicated situations involving collinearity, but in this case simply correlating the two predictors should establish the basic point.
I would like to fit a regression model to probabilities. I am aware that linear regression is often used for this purpose, but I have several probabilities at or near 0.0 and 1.0 and would like to fit a regression model where the output is constrained to lie between 0.0 and 1.0. I want to be able to specify a regularization norm and strength for the model and ideally do this in python (but an R implementation would be helpful as well). All the logistic regression packages I've found seem to be only suited for classification whereas this is a regression problem (albeit one where I want to use the logit link function). I use scikits-learn for my classification and regression needs so if this regression model can be implemented in scikits-learn, that would be fantastic (it seemed to me that this is not possible), but I'd be happy about any solution in python and/or R.
The question has two issues, penalized estimation and fractional or proportions data as dependent variable. I worked on each separately but never tried the combination.
Penalization
Statsmodels has had L1 regularized Logit and other discrete models like Poisson for some time. In recent months there has been a lot of effort to support more penalization but it is not in statsmodels yet. Elastic net for linear and Generalized Linear Model (GLM) is in a pull request and will be merged soon. More penalized GLM like L2 penalization for GAM and splines or SCAD penalization will follow over the next months based on pull requests that still need work.
Two examples for the current L1 fit_regularized for Logit are here
Difference in SGD classifier results and statsmodels results for logistic with l1 and https://github.com/statsmodels/statsmodels/blob/master/statsmodels/examples/l1_demo/short_demo.py
Note, the penalization weight alpha can be a vector with zeros for coefficients like the constant if they should not be penalized.
http://www.statsmodels.org/dev/generated/statsmodels.discrete.discrete_model.Logit.fit_regularized.html
Fractional models
Binary and binomial models in statsmodels do not impose that the dependent variable is binary and work as long as the dependent variable is in the [0,1] interval.
Fractions or proportions can be estimated with Logit as Quasi-maximum likelihood estimator. The estimates are consistent if the mean function, logistic, cumulative normal or similar link function, is correctly specified but we should use robust sandwich covariance for proper inference. Robust standard errors can be obtained in statsmodels through a fit keyword cov_type='HC0'.
Best documentation is for Stata http://www.stata.com/manuals14/rfracreg.pdf and the references therein. I went through those references before Stata had fracreg, and it works correctly with at least Logit and Probit which were my test cases. (I don't find my scripts or test cases right now.)
The bad news for inference is that robust covariance matrices have not been added to fit_regularized, so the correct sandwich covariance is not directly available. The standard covariance matrix and standard errors of the parameter estimates are derived under the assumption that the model, i.e. the likelihood function, is correctly specified, which will not be the case if the data are fractions and not binary.
Besides using Quasi-Maximum Likelihood with binary models, it is also possible to use a likelihood that is defined for fractional data in (0, 1). A popular model is Beta regression, which is also waiting in a pull request for statsmodels and is expected to be merged within the next months.
How should one decide between using a linear regression model or non-linear regression model?
My goal is to predict Y.
In case of simple x and y dataset I could easily decide which regression model should be used by plotting a scatter plot.
In case of multi-variant like x1,x2,...,xn and y. How can I decide which regression model has to be used? That is, How will I decide about going with simple linear model or non linear models such as quadric, cubic etc.
Is there any technique or statistical approach or graphical plots to infer and decide which regression model has to be used? Please advise.
That is a pretty complex question.
You start visually first: if the data is normally distributed, and satisfy conditions for classical linear model, you use linear model. I normally start by making a scatter plot matrix to observe the relationships. If it is obvious that the relationship is non linear then you use non-linear model. But, a lot of times, I visually inspect, assuming that the number of factors are just not too many.
For example, this would be a non linear model:
However, if you want to use data mining (and computationally demanding methods), I suggest starting with stepwise regression. What you do is set a model evaluation criteria first: could be R^2 for example. You start a model with nothing and sequentially add predictors or permutations of them until your model evaluation criteria is "maximized". However, adding new predictor almost always increases R^2, a type of over-fitting.
The solution is to split the data into training and testing. You should make model based on the training and evaluate the mean error on testing. The best model will be the one that that minimized mean error on the testing set.
If your data is sparse, try integrating ridge or lasso regression in model evaluation.
Again, this is a kind of a complex question. The answer also kind of depends on whether you are building descriptive or explanatory model.
I have data which has an associated binary outcome variable. Naturally I ran a logistic regression in order to see parameter estimates and odds ratios. I was curious though, to change this data from a binary outcome to count data. Then I ran a poisson regression (and negative binomial regression) on the count data.
I have no idea of how to compare these different models though, all comparisons I see seem to only be concerned with nested models.
How would you go about deciding on the best model to use in this situation?
Essentially both models will be roughly equal. What really matters is what is your objective- what you really want to predict. If you want to determine how many of cases are good or bad (1 or 0), then you go for logistic regression. If you are really interested on how much the cases are going to do (counts) then do poisson.
In other words, the only difference between these two models is the logistic transformation and the fact that logistic regression tries to minimize the misclassification error (-2 log likelihood) .To put it simply, even if you run a linear regression (OLS) on the binary outcome, you should not see big differences from your logistic model apart from the fact that the results may not be between 0 and 1 (e.g. the Area under the RoC curve will be similar to the logistic model) .
To sum up, don't worry about which of these two models is better, they should be roughly the same in the way the capture your features' information. Just think what makes more sense to optimize, counts or probabilties. The answer might have been different if you were considering non-linear models (e.g random forests or neural networks etc), but the two you are considering are both (almost) linear- so don't worry about it.
One thing to consider is the sample design. If you are using a case-control study, then logistic regression is the way to go because of its logit link function, rather than log of ratios as in Poisson regression. This is because, where there is an oversampling of cases such as in case-control study, odds ratio is unbiased.