I am fitting a standard multiple regression with OLS method. I have 5 predictors (2 continuous and 3 categorical) plus 2 two-way interaction terms. I did regression diagnostics using residuals vs. fitted plot. Heteroscedasticity is quite evident, which is also confirmed by bptest().
I don't know what to do next. First, my dependent variable is reasonably symmetric (I don't think I need to try transformations of my DV). My continuous predictors are also not highly skewed. I want to use weights in lm(); however, how do I know what weights to use?
Is there a way to automatically generate weights for performing weighted least squares? or Are you other ways to go about it?
One obvious way to deal with heteroscedasticity is the estimation of heteroscedasticity consistent standard errors. Most often they are referred to as robust or white standard errors.
You can obtain robust standard errors in R in several ways. The following page describes one possible and simple way to obtain robust standard errors in R:
https://economictheoryblog.com/2016/08/08/robust-standard-errors-in-r
However, sometimes there are more subtle and often more precise ways to deal with heteroscedasticity. For instance, you might encounter grouped data and find yourself in a situation where standard errors are heterogeneous in your dataset, but homogenous within groups (clusters). In this case you might want to apply clustered standard errors. See the following link to calculate clustered standard errors in R:
https://economictheoryblog.com/2016/12/13/clustered-standard-errors-in-r
What is your sample size? I would suggest that you make your standard errors robust to heteroskedasticity, but that you do not worry about heteroskedasticity otherwise. The reason is that with or without heteroskedasticity, your parameter estimates are unbiased (i.e. they are fine as they are). The only thing that is affected (in linear models!) is the variance-covariance matrix, i.e. the standard errors of your parameter estimates will be affected. Unless you only care about prediction, adjusting the standard errors to be robust to heteroskedasticity should be enough.
See e.g. here how to do this in R.
Btw, for your solution with weights (which is not what I would recommend), you may want to look into ?gls from the nlme package.
Related
Unfortunately, I had convergence (and singularity) issues when calculating my GLMM analysis models in R. When I tried it in SPSS, I got no such warning message and the results are only slightly different. Does it mean I can interpret the results from SPSS without worries? Or do I have to test for singularity/convergence issues to be sure?
You have two questions. I will answer both.
First Question
Does it mean I can interpret the results from SPSS without worries?
You do not want to do this. The reason being is that mixed models have a very specific parameterization. Here is a screenshot of common lme4 syntax from the original article about lme4 from the author:
With this comes assumptions about what your model is saying. If for example you are running a model with random intercepts only, you are assuming that the slopes do not vary by any measure. If you include correlated random slopes and random intercepts, you are then assuming that there is a relationship between the slopes and intercepts that may either be positive or negative. If you present this data as-is without knowing why it produced this summary, you may fail to explain your data in an accurate way.
The reason as highlighted by one of the comments is that SPSS runs off defaults whereas R requires explicit parameters for the model. I'm not surprised that the model failed to converge in R but not SPSS given that SPSS assumes no correlation between random slopes and intercepts. This kind of model is more likely to converge compared to a correlated model because the constraints that allow data to fit a correlated model make it very difficult to converge. However, without knowing how you modeled your data, it is impossible to actually know what the differences are. Perhaps if you provide an edit to your question that can be answered more directly, but just know that SPSS and R do not calculate these models the same way.
Second Question
Or do I have to test for singularity/convergence issues to be sure?
SPSS and R both have singularity checks as a default (check this page as an example). If your model fails to converge, you should drop it and use an alternative model (usually something that has a simpler random effects structure or improved optimization).
Working with a dataset of ~200 observations and a number of variables. Unfortunately, none of the variables are distributed normally. If it possible to extract a data subset where at least one desired variable will be distributed normally? Want to do some statistics after (at least logistic regression).
Any help will be much appreciated,
Phil
If there are just a few observations that skew the distribution of individual variables, and no other reasons speaking against using a particular method (such as logistic regression) on your data, you might want to study the nature of "weird" observations before deciding on which analysis method to use eventually.
I would:
carry out the desired regression analysis (e.g. logistic regression), and as it's always required, carry out residual analysis (Q-Q Normal plot, Tukey-Anscombe plot, Leverage plot, also see here) to check the model assumptions. See whether the residuals are normally distributed (the normal distribution of model residuals is the actual assumption in linear regression, not that each variable is normally distributed, of course you might have e.g. bimodally distributed data if there are differences between groups), see if there are observations which could be regarded as outliers, study them (see e.g. here), and if possible remove them from the final dataset before re-fitting the linear model without outliers.
However, you always have to state which observations were removed, and on what grounds. Maybe the outliers can be explained as errors in data collection?
The issue of whether it's a good idea to remove outliers, or a better idea to use robust methods was discussed here.
as suggested by GuedesBF, you may want to find a test or model method which has no assumption of normality.
Before modelling anything or removing any data, I would always plot the data by treatment / outcome groups, and inspect the presence of missing values. After quickly looking at your dataset, it seems that quite some variables have high levels of missingness, and your variable 15 has a lot of zeros. This can be quite problematic for e.g. linear regression.
Understanding and describing your data in a model-free way (with clever plots, e.g. using ggplot2 and multiple aesthetics) is much better than fitting a model and interpreting p-values when violating model assumptions.
A good start to get an overview of all data, their distribution and pairwise correlation (and if you don't have more than around 20 variables) is to use the psych library and pairs.panels.
dat <- read.delim("~/Downloads/dput.txt", header = F)
library(psych)
psych::pairs.panels(dat[,1:12])
psych::pairs.panels(dat[,13:23])
You can then quickly see the distribution of each variable, and the presence of correlations among each pair of variables. You can tune arguments of that function to use different correlation methods, and different displays. Happy exploratory data analysis :)
I'm modelling multiple linear regression. I used the bptest function to test for heteroscedasticity. The result was significant at less than 0.05.
How can I resolve the issue of heteroscedasticity?
Try using a different type of linear regression
Ordinary Least Squares (OLS) for homoscedasticity.
Weighted Least Squares (WLS) for heteroscedasticity without correlated errors.
Generalized Least Squares (GLS) for heteroscedasticity with correlated errors.
Welcome to SO, Arun.
Personally, I don't think heteroskedasticity is something you "solve". Rather, it's something you need to allow for in your model.
You haven't given us any of your data, so let's assume that the variance of your residuals increases with the magnitude of your predictor. Typically a simplistic approach to handling it is to transform the data so that the variance is constant. One way of doing this might be to log-transform your data. That might give you a more constant variance. But it also transforms your model. Your errors are no longer IID.
Alternatively, you might have two groups of observarions that you want to compare with a t-test, bit the variance in one group is larger than in the other. That's a different sot of heteroskedasticity. There are variants of the standard "pooled variance" t-test that might handle that.
I realise this isn't an answer to your question in the conventional sense. I would have made it a comment, but I knew before I started that I'd need more words than a comment would let me have.
I am using R 3.2.0 with lme4 version 1.1.8. to run a mixed effects logistic regression model on some binomial data (coded as 0 and 1) from a psycholinguistic experiment. There are 2 categorical predictors (one with 2 levels and one with 3 levels) and two random terms (participants and items). I am using sum coding for the predictors (i.e. contr.sum..) which gives me the effects and interactions that I am interested in.
I find that the full model (with fixed effects and interactions, plus random intercepts AND slopes for the two random terms) converges ONLY when I specify (optimizer="bobyqa"). If I do not specify the optimizer, the model converges only after simplifying the model drastically. The same thing happens when I use the default treatment coding, even when I specify optimizer="bobyqa".
My first question is why is this happening and can I trust the output of the full model?
My second question is whether this might be due to the fact that my data is not fully balanced, in the sense that my conditions do not have exactly the same number of observations. Are there special precautions one must take when the data is not full balanced? Can one suggest any reading on this particular case?
Many thanks
You should take a look at the ?convergence help page of more recent versions of lme4 (or you can read it here). If the two fits using different optimizers give similar estimated parameters (despite one giving convergence warnings and the other not), and the fits with different contrasts give the same log-likelihood, then you probably have a reasonable fit.
In general lack of balance lowers statistical power and makes fitting more difficult, but mildly to moderate unbalanced data should present no particular problems.
I want to compare the curve fits of three models by r-squared values. I ran models using the nls and drc packages. It appears, though, that neither of those packages calculate r-squared values; they give "residual std error" and "residual sum of squares" though.
Can these two be used to compare model fits?
This is really a statistics question, rather than a coding question: consider posting on stats.stackexchange.com; you're likely to get a better answer.
RSQ is not really meaningful for non-linear regression. This is why summary.nls(...) does not provide it. See this post for an explanation.
There is a common, and understandable, tendency to hope for a single statistic that allows one to assess which of a set of models better fits a dataset. Unfortunately, it doesn't work that way. Here are some things to consider.
Generally, the best model is the one that has a mechanistic underpinning. Do your models reflect some physical process, or are you just trying a bunch of mathematical equations and hoping for the best? The former approach almost always leads to better models.
You should consider how the models will be used. Will you be interpolating (e.g. estimating y|x within the range of your dataset), or will you be extrapolating (estimating y|x outside the range of your data)? Some models yield a fit that provides relatively accurate estimates slightly outside the dataset range, and others completely fall apart.
Sometimes the appropriate modeling technique is suggested by the type of data you have. For example, if you have data that counts something, then y is likely to be poisson distributed and a generalized linear model (glm) in the poisson family is indicated. If your data is binary (e.g. only two possible outcomes, success or failure), then a binomial glm is indicated (so-called logistic regression).
The key underlying assumption of least squares techniques is that the error in y is normally distributed with mean 0 and constant variance. We can test this after doing the fit by looking at a plot of standardized residuals vs. y, and by looking at a Normal Q-Q plot of the residuals. If the residuals plot shows scatter increasing or decreasing with y then the model in not a good one. If the Normal Q-Q plot is not close to a straight line, then the residuals are not normally distributed and probably a different model is indicated.
Sometimes certain data points have high leverage with a given model, meaning that the fit is unduly influenced by those points. If this is a problem you will see it in a leverage plot. This indicates a weak model.
For a given model, it may be the case that not all of the parameters are significantly different from 0 (e.g., p-value of the coefficient > 0.05). If this is the case, you need to explore the model without those parameters. With nls, this often implies a completely different model.
Assuming that your model passes the tests above, it is reasonable to look at the F-statistic for the fit. This is essentially the ratio of SSR/SSE corrected for the dof in the regression (R) and the residuals (E). A model with more parameters will generally have smaller residual SS, but that does not make it a better model. The F-statistic accounts for this in that models with more parameters will have larger regression dof and smaller residual dof, making the F-statistic smaller.
Finally, having considered the items above, you can consider the residual standard error. Generally, all other things being equal, smaller residual standard error is better. Trouble is, all other things are never equal. This is why I would recommend looking at RSE last.