# How to interpret regression coefficients when each predictor variable contains different categories

Overview:

I have conducted two types of statistical analysis using both linear regression and multiple regression.

Overall, there were two observation periods, and the idea is to gauge if the rate of tree senescence (trees losing leaves in autumn) has significantly been impacted by various different parameters (listed below) between both observation periods.

The independent variables all contain different categories:

Key:

1. Urbanisation Index: 1=Urban, 2=Suburban, 3=village, 4=rural

2. Stand Density Index: 1=standing alone, 2=within a few trees or close proximity to other trees, 3=within a stand of 10-30 trees, and 4=large or woodland

3. Phenological Index: 1=no indication of autumn timing, 2=first autumn tinting, 3=partial autumn tinting (>25% of leaves), and 4=advanced autumn tinting (>75% of leaves)

Question 1:

1. Using the full database and the results from an appropriate statistical test, accept or reject the following hypothesis at the 5 % significance level.

H(0): There is no difference in stem diameter of Quercus robur between the different categories of stand density index.

Question 2

Discuss whether the factors of urbanisaiton and stand density influence leaf senescence (losing leaves), as measured by the phenological index, in Quercus robur. Based on your results, comment on whether you think urbanisation or stand density are the source of influence on leaf senescence in Quercus robur, and whether one factor has a stronger influence than the other.

I am feeling confused with how to interpret the table of coefficients produced by the summary statistics for all regression models.

If anyone can advise to clear the confusion, I would be deeply appreciative.

Issues:

1. For question 1, I used a linear regression model (see R-code below) with y=Tree diameter and x=Stand Density Index. From my understanding, the summary statistics denotes tree stem diameter is significantly different for category 1 or the intercept in the stand density index (stand alone trees) or observation 1 (p=0.00979) observation 2 (p=0.00808) with a significance level of P < 0.01.with a significance level of P < 0.01, in comparison to categories 2-4, which are insignificant. Am I interpreting these coefficients correctly?

2. For question 2 in the multiple regression analysis, would I be correct in saying that category 1 or the intercept of the stand density index (standing alone trees), and category 4 of the urbansiation index (rural locations) significantly affected the rate of phenological change in leaf senescence shown in observation 1 (p=2e-16, Stand Density Category 1; p=0.0492, Urbanisation Index 4) and observation 2 ((1) Stand Density Category 1, p=2e-16; (2) Urbanisation Index 2, p=0.01485 (3) Urbanisation Index 3, p=0.0527; and 4. Urbanisation index 4, p=0.0225).Am I interpreting these coefficients correctly?

3. I am confused as to why category 1 for the Urbanisation Index and stand density index is not visible in the table of coefficients for both multiple regression models for both observation 1 and observation 2.Has category 1 for both parameters been integrated into the summary statistic for intercept in order to make direct comparisons with the other categories for both predictors?

4. In order to find the strongest factor affecting the phenological index, I used the caret package and the function varImp() to sort the most important predictor variables by their relative importance regarding the percentage (%) of contribution to the model. According to the results, category 2 of the stand density index (within a few trees or close proximity to other trees) is the most important predictor for both observation 1 (81.0 %) and observation 2 (77.3 %).Is this is the correct method?

Linear Regression: Question 1

        ##Reformat stand_density_index into a categorical vector
QuercusRobur1$$Stand_density_index<-as.factor(QuercusRobur1$$Stand_density_index)
QuercusRobur2$$Stand_density_.index<-as.factor(QuercusRobur2$$Stand_density_.index)

##Linear Regression
##Observation 1
StemDensityStand1<-lm(Tree_diameter~Stand_density_index, data=QuercusRobur1)
summary(StemDensityStand1)

##Intercept = 0 model: Observation 1
summary(lm(Tree_diameter~0+Stand_density_index, data=QuercusRobur1))

##Observation 2
StemDensityStand2<-lm(Tree_diameter~Stand_density_.index, data=QuercusRobur2)
summary(StemDensityStand2)

##Intercept = 0 model: Observation 1
summary(lm(Tree_diameter~0+Stand_density_.index, data=QuercusRobur2))


Results:

Multiple Regression: Question 2

############Reformat Urbansiation Index into a factor

QuercusRobur1$$Urbanisation_index<-as.factor(QuercusRobur1$$Urbanisation_index)
QuercusRobur2$$Urbanisation_index<-as.factor(QuercusRobur2$$Urbanisation_index)
##############Multiple Regression

MultiplePhenStandUrban1<-lm(Phenological_Index~Stand_density_index+Urbanisation_index, data=QuercusRobur1)
MultiplePhenStandUrban2<-lm(Phenological_Index~Stand_density_.index+Urbanisation_index, data=QuercusRobur2)

##############Summary Statistics
[![enter image description here]]          summary(MultiplePhenStandUrban1)
summary(MultiplePhenStandUrban2)


Results

What are the most important predictors in the multiple regression models?

library(caret)

##Observation 1
varImp(MultiplePhenStandUrban1, scale=FALSE)

##Observation 2
varImp(MultiplePhenStandUrban2, scale=FALSE)


Results: 