# Interpreting estimates of a bivariate regression model with a categotical and a numeric variable

How to interpret the intercept in a bivariate regression with one numeric and one categorical variable?

The following model has a numeric variable (log10(N_Total_e)), namely the log transformed amount of nitrogen combined with a categorical variable divided in three levels of fertilizer class (NH4 only, NO3 only, NH4+NO3 combined). Dependent variable is species richness.

Multivariate Meta-Analysis Model (k = 211; method: REML)

logLik  Deviance       AIC       BIC      AICc
-33.0673   66.1345   76.1345   92.7981   76.4331

Variance Components:

estim    sqrt  nlvls  fixed         factor
sigma^2    0.0125  0.1116     73     no  Primary_Study

Test for Residual Heterogeneity:
QE(df = 207) = 132.2270, p-val = 1.0000

Test of Moderators (coefficient(s) 2,3,4):
QM(df = 3) = 13.9606, p-val = 0.0030

Model Results:

estimate      se     zval    pval    ci.lb    ci.ub
intrcpt             1.2338  0.1358   9.0846  <.0001   0.9676   1.5000  ***
log10(N_Total_e)   -0.1938  0.0667  -2.9071  0.0036  -0.3245  -0.0632   **
F_typeNH4+NO3      -0.1328  0.0579  -2.2952  0.0217  -0.2463  -0.0194    *
F_typeNO3          -0.1221  0.1368  -0.8927  0.3720  -0.3903   0.1460


Is it right to say that the amount of nitrogen (log10(N_Total_e)) has in general a negative effect species richness (estimate = -0.19)? Should I interpret F_typeNH4+NO3 and F_typeNO3 estimates equals to 1.10 and 1.11 (namely the sum of the estimate with the intercept)? So specifying the type of fertilizer will always increase species richness when one is making predictions according to this model. Is this interpretation correct?

• Note that you can get an omnibus test of Ftype by using the btt parameter in rma.mv – mdewey Nov 7 '16 at 14:23
• You might also benefit from reading this Q&A stats.stackexchange.com/questions/242956/… especially @Wolfgang's suggestion about a three level model. – mdewey Nov 8 '16 at 14:26

## 1 Answer

Since you are using dummy coding to handle the categorical variable, your intercept and the regression coefficients for the dummy codes will reference the group you left out of the dummy coding (NH4 only). The intercept is the species richness with NH4 only and a nitrogen log value of $0.0$, and its p-value describes if this value is different from zero. The regression coefficient for each dummy code is the difference between that level of the categorical variable and the reference group (NH4 only).

You are correct that you always need to add in the intercept. So species richness with NH4 only and a nitrogen log value of $0.0$ is $1.2338$. And when the fertilizer type is N03 with a nitrogen log value of $0.0$, species richness equals $1.2338-0.1228=1.111$. For every log unit increase in nitrogen, species richness decreases by another $0.1938$.

Each type of fertilizer increases the species richness, but to different degrees. The NH4 only fertilizer had the highest amount of species richness and this value was significantly higher than that of NH4+N03. The difference between NH4 and N03 was not significantly different. To see the contrast of N03 and NH4+N03, you would need to run the model again with a different reference group.

• So is it correct to say that the values of fertilizer type represent the value of the intercept that the function assumes if "NH4 only", "NH4+NO3" or "NO3 only" are used (1.23, 1.10, 1.11 respectively)? Then the estimate for the amount of N is just the slope (i.e. the decrease in species richness for an increase of 1 of log10(N_Total_e)), but the intercept value depend on the type of fertilizer used. I did not get fully your point when you say that fertilizer increases species richness. – Gabriele Midolo Nov 7 '16 at 15:56
• Yeah, that's correct. What I meant by "each type of fertilizer increases the species richness" is that all of the intercepts + coefficients are positive numbers, so the relationship is always positive. It's not like NO3 reduces species richness, it's just a smaller increase than that of NH4+NO3. – Jeffrey Girard Nov 8 '16 at 1:39