Solved – How to interpret regression coefficients for a variable with takes positive and negative values

generalized-estimating-equationsregressionregression coefficients

I am running a GEE negative binomial regression to see how predictors affect the onset of violence through time.

I have an $X$ variable (vegetation cover) which is calculated as whether an observation differed positively or negatively from the mean of the overall $X$ variable. It was calculated from raw values and covers a 10 year period with each observation unit differing (positively or negatively) from the 10 year mean of all summed raw obervations.

How do I interpret my $\beta$ coefficients if my sig values are significant for such a variable? Does it tell me if a specific positive or negative change in vegetation is significant, or just that a change from the mean generally is significant?

Best Answer

If I understand you correctly you have mean centered your independent variable by subtracting the mean value of that variable from all observations.

If so, then the coefficient reflects the effect of a 1 unit increase in your independent variable, just as it would if the variable were uncentered. What centering does is change the interpretation of the intercept/constant as well as any interaction terms involved with the centered variable. Many seem to think that centering resolves collinearity problems when it certainly does not but instead merely masks them by shifting the collinearity onto the intercept.

Try running the model with the uncentered term and the centered term. Do you see any difference in the effect for that variable?

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Solved – Regression with autocorrelated, lagged independent variable

Ordinary Least Squares estimation is inadequate in the existence of heteroscedasticity- unequal and out-of-diagonal covariance terms among the observations.

In your example, LST is a time-series which you assume to have autoregressive nature. Therefore, when using successive terms from this time series as regressors, you should be aware the OLS may be inefficient and yield over-optimistic error estimates. Two methods that address these issues are:

Generalized Least Squares - This approach employs the observation correlation matrix (or in most practical cases, an estimate of the correlation matrix) and can be intuitively explained as equivalent to performing the regression in a transformed vector space where the observations are uncorrelated.
OLS with robust variance estimation - With this approach, the regression coefficients (B in your example) are calculated the same way as with OLS, but the error terms' variance is estimated in any of the available techniques that make the estimation more robust to the existence of correlations and unequal variances.

Googling "Generalized least squares" or "robust variance estimation" should get you useful resources. In addition, see Gretl User's Guide, chapter 17.

Regression – Negative Regression Coefficient Despite Positive Raw Plot

What is happening here is essentially just Simpson's "paradox". In this particular case you have observed positive marginal correlation between yield and x1, but the relationship turns negative after you condition on x2 and year in your linear model. You can also see from your plots that x1 and x2 have strong positive correlation, so this is giving you strong multicollinearity which would explain the phenomenon in this case.

This type of phenomenon is not unusual when examining relationships between multiple variables, especially when there is strong collinearity. For this reason it is generally misleading to plot crude pairwise comparisons between variables when doing analysis with many variables. If you want to look at the conditional relationship between yield and x1 then this would usually be illustrated with an partial regression plot (also called an added variable plot).

Implementation in R: The effects package has functionality to automatically produce residuals that absorb the lower-order terms marginal to the model variable of interest. This allows you to construct what are effectively partial regression plots for a range of models including lme models. This can be implemented to produce a partial regression plot in R using the code below. (Note that the data file you have linked to does not exactly match with the model output you have presented in your question. I have included the model output from the linked data.)

#Read data (need to put it in working directory first)
DATA <- read.csv('my.data.csv');

#Fit your model
library(lme4);
MODEL <- lmer(log(yield) ~ x1 + x2 + year + (year |admin1/admin2),
              REML = FALSE, data = DATA);

#Show model output
summary(MODEL);

...
Fixed effects:
            Estimate Std. Error t value
(Intercept)  8.41434    0.08585  98.008
x1          -0.07381    0.01558  -4.736
x2           0.13214    0.01521   8.687
year         0.11642    0.02994   3.888
....

#Generate partial regression plot using effects package
library(effects);
PARTIAL_MODEL <- Effect('x1', partial.residuals = TRUE, mod = MODEL);
plot(PARTIAL_MODEL, main = 'Partial Regression Plot',
     xlab = 'x1', ylab = 'Log-Yield');

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Related Solutions

Solved – Regression with autocorrelated, lagged independent variable

Regression – Negative Regression Coefficient Despite Positive Raw Plot

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