Project Analysis

1.1 Factors: Canopy cover percent (continuous) ~ Ash dieback levels (categories)

#check normality of data 
library(dplyr)
library(ggpubr)
library(car)
#change the colname of Canopy_cover_percent in Ash data
colnames(Ash)[colnames(Ash)=="Canopy_cover_percent"] <- "Canopy_cover"
colnames(Ash)[colnames(Ash)=="Ash_dieback"] <- "AD"
#the qq plot shows characteristic of binomial data
p <- ggqqplot(Ash$Canopy_cover)
p

#change the factor levels of AD
Ash$AD <- factor(Ash$AD, levels = c("low", "medium", "high"))
#boxplot
p <- ggboxplot(Ash, x="AD", y= "Canopy_cover",color = "AD", palette = c("#00AFBB", "#E7B800", "#FC4E07")) + labs(x="Extent of ash tree dieback", y = "Canopy Cover (%)") + geom_jitter(shape=19, position=position_jitter(0.2)) + geom_point() + theme(legend.position = "none")
p

#GLM of all three factors
library(MASS)
Canopy_AD <- glm(cbind(Canopy_cover,Canopy_not_cover) ~ AD, data=Ash, family=binomial)

non-integer counts in a binomial glm!

summary(Canopy_AD)

Call:
glm(formula = cbind(Canopy_cover, Canopy_not_cover) ~ AD, family = binomial, 
    data = Ash)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-3.8170  -1.0968  -0.1488   1.4751   3.3650  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -0.41139    0.06808  -6.043 1.52e-09 ***
ADmedium    -0.10631    0.09439  -1.126     0.26    
ADhigh      -1.04340    0.09639 -10.825  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 271.00  on 32  degrees of freedom
Residual deviance: 117.65  on 30  degrees of freedom
AIC: 280.69

Number of Fisher Scoring iterations: 4

step.model1 <- stepAIC(Canopy_AD, direction = "backward", 
                       trace = FALSE)

non-integer #successes in a binomial glm!

summary(step.model1)

Call:
glm(formula = cbind(Canopy_cover, Canopy_not_cover) ~ AD, family = binomial, 
    data = Ash)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-3.8170  -1.0968  -0.1488   1.4751   3.3650  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -0.41139    0.06808  -6.043 1.52e-09 ***
ADmedium    -0.10631    0.09439  -1.126     0.26    
ADhigh      -1.04340    0.09639 -10.825  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 271.00  on 32  degrees of freedom
Residual deviance: 117.65  on 30  degrees of freedom
AIC: 280.69

Number of Fisher Scoring iterations: 4

Canopy_AD_BA <- glm(cbind(Canopy_cover,Canopy_not_cover) ~ AD * Understorey, data=Ash, family=binomial)

non-integer counts in a binomial glm!

summary(Canopy_AD_BA)

Call:
glm(formula = cbind(Canopy_cover, Canopy_not_cover) ~ AD * Understorey, 
    family = binomial, data = Ash)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-3.8910  -1.1736  -0.1596   1.4570   3.3494  

Coefficients:
                      Estimate Std. Error z value Pr(>|z|)    
(Intercept)           -0.49178    0.10304  -4.773 1.82e-06 ***
ADmedium              -0.09076    0.17989  -0.504    0.614    
ADhigh                -0.96580    0.13703  -7.048 1.81e-12 ***
Understorey+           0.14375    0.13734   1.047    0.295    
ADmedium:Understorey+ -0.06288    0.21431  -0.293    0.769    
ADhigh:Understorey+   -0.13726    0.19457  -0.705    0.481    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 271.00  on 32  degrees of freedom
Residual deviance: 116.31  on 27  degrees of freedom
AIC: 285.13

Number of Fisher Scoring iterations: 4

step.model2 <- stepAIC(Canopy_AD_BA, direction = "backward", 
                       trace = FALSE)

non-integer #successes in a binomial glm!non-integer counts in a binomial glm!non-integer #successes in a binomial glm!non-integer #successes in a binomial glm!non-integer counts in a binomial glm!non-integer #successes in a binomial glm!

summary(step.model2)

Call:
glm(formula = cbind(Canopy_cover, Canopy_not_cover) ~ AD, family = binomial, 
    data = Ash)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-3.8170  -1.0968  -0.1488   1.4751   3.3650  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -0.41139    0.06808  -6.043 1.52e-09 ***
ADmedium    -0.10631    0.09439  -1.126     0.26    
ADhigh      -1.04340    0.09639 -10.825  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 271.00  on 32  degrees of freedom
Residual deviance: 117.65  on 30  degrees of freedom
AIC: 280.69

Number of Fisher Scoring iterations: 4

dropterm(step.model2,test = "Chisq")

non-integer #successes in a binomial glm!

Single term deletions

Model:
cbind(Canopy_cover, Canopy_not_cover) ~ AD
       Df Deviance    AIC    LRT   Pr(Chi)    
<none>      117.65 280.69                     
AD      2   271.00 430.05 153.35 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#if dropped AD:BA, AIC increases, so don't drop
Canopy_AD_BA_2 <- glm(cbind(Canopy_cover,Canopy_not_cover) ~ AD + Understorey, data=Ash, family=binomial)

non-integer counts in a binomial glm!

anova(Canopy_AD_BA, Canopy_AD_BA_2, test="Chisq")

Analysis of Deviance Table

Model 1: cbind(Canopy_cover, Canopy_not_cover) ~ AD * Understorey
Model 2: cbind(Canopy_cover, Canopy_not_cover) ~ AD + Understorey
  Resid. Df Resid. Dev Df Deviance Pr(>Chi)
1        27     116.31                     
2        29     116.81 -2 -0.49918   0.7791

summary(Canopy_AD_BA_2)

Call:
glm(formula = cbind(Canopy_cover, Canopy_not_cover) ~ AD + Understorey, 
    family = binomial, data = Ash)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-3.8873  -1.1698  -0.1798   1.6026   3.1812  

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept)  -0.45418    0.08261  -5.498 3.84e-08 ***
ADmedium     -0.12504    0.09658  -1.295    0.195    
ADhigh       -1.03396    0.09691 -10.669  < 2e-16 ***
Understorey+  0.07675    0.08361   0.918    0.359    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 271.00  on 32  degrees of freedom
Residual deviance: 116.81  on 29  degrees of freedom
AIC: 281.82

Number of Fisher Scoring iterations: 4

anova(Canopy_AD_BA,Canopy_AD_BA_2)

Analysis of Deviance Table

Model 1: cbind(Canopy_cover, Canopy_not_cover) ~ AD * Understorey
Model 2: cbind(Canopy_cover, Canopy_not_cover) ~ AD + Understorey
  Resid. Df Resid. Dev Df Deviance
1        27     116.31            
2        29     116.81 -2 -0.49918

#therefore understorey and the interaction between AD and understorey are not important

Design a linear regression plot function

ggplotRegression <- function (fit) {
    
    require(ggplot2)
    
    ggplot(fit$model, aes_string(x = names(fit$model)[2], y = names(fit$model)[1])) + 
        geom_point() +
        stat_smooth(method = "lm", col = "red") +
        labs(title = paste("Adj R2 = ",signif(summary(fit)$adj.r.squared, 5),
                           "Intercept =",signif(fit$coef[[1]],5 ),
                           " Slope =",signif(fit$coef[[2]], 5),
                           " P =",signif(summary(fit)$coef[2,4], 5)))
}

Factor 2: temperature fluctuation vs AD & LAI

## Add temperature summaries to Ash data
Ash$Max_temp <- Temp_data$Max_temp[match(Ash$Site, Temp_data$Site)]
Ash$Min_temp <- Temp_data$Min_temp[match(Ash$Site, Temp_data$Site)]
Ash$Temp_range <- Temp_data$Temp_range[match(Ash$Site, Temp_data$Site)] 
#Max temp vs AD
#test normality (not normal, SW test: p = 0.036)
p <- ggqqplot(Ash$Max_temp)
p

shapiro.test(Ash$Max_temp)

    Shapiro-Wilk normality test

data:  Ash$Max_temp
W = 0.93058, p-value = 0.03634

#boxplot of AD ~ Max temp
p <- ggboxplot(Ash, x="AD", y= "Max_temp",color = "AD", palette = c("#00AFBB", "#E7B800", "#FC4E07")) + labs(x="Extent of ash tree disback", y = "Maximum temerature (°C)") + geom_jitter(shape=19, position=position_jitter(0.2)) + geom_point() + theme(legend.position = "none")
p

#Wilcoxon rank sum test: 
kruskal.test(Max_temp ~ AD, data = Ash)

    Kruskal-Wallis rank sum test

data:  Max_temp by AD
Kruskal-Wallis chi-squared = 13.316, df = 2, p-value = 0.001283

pairwise.wilcox.test(Ash$Max_temp, Ash$AD, p.adjust.method = "BH")

cannot compute exact p-value with tiescannot compute exact p-value with tiescannot compute exact p-value with ties

    Pairwise comparisons using Wilcoxon rank sum test 

data:  Ash$Max_temp and Ash$AD 

       low    medium
medium 0.9020 -     
high   0.0045 0.0045

P value adjustment method: BH 

#significance is between medium and high levels of AD (p=0.0045)
#Minimum Temp vs AD
p <- ggqqplot(Ash$Min_temp)
p

shapiro.test(Ash$Min_temp) #not normal again

    Shapiro-Wilk normality test

data:  Ash$Min_temp
W = 0.93681, p-value = 0.05489

p <- ggboxplot(Ash, x="AD", y= "Min_temp",color = "AD", palette = c("#00AFBB", "#E7B800", "#FC4E07")) + labs(x="Extent of ash tree disback", y = "Minimum temerature (°C)") + geom_jitter(shape=19, position=position_jitter(0.2)) + geom_point() + theme(legend.position = "none")
p

#Wilcoxon rank sum test: 
kruskal.test(Min_temp ~ AD, data = Ash)

    Kruskal-Wallis rank sum test

data:  Min_temp by AD
Kruskal-Wallis chi-squared = 4.9176, df = 2, p-value = 0.08554

pairwise.wilcox.test(Ash$Min_temp, Ash$AD, p.adjust.method = "BH")

cannot compute exact p-value with tiescannot compute exact p-value with tiescannot compute exact p-value with ties

    Pairwise comparisons using Wilcoxon rank sum test 

data:  Ash$Min_temp and Ash$AD 

       low  medium
medium 0.62 -     
high   0.16 0.16  

P value adjustment method: BH 

#no significance 
#Temp range vs AD
p <- ggqqplot(Ash$Temp_range)
p

shapiro.test(Ash$Temp_range) #not very normal again

    Shapiro-Wilk normality test

data:  Ash$Temp_range
W = 0.93922, p-value = 0.0645

p <- ggboxplot(Ash, x="AD", y= "Temp_range",color = "AD", palette = c("#00AFBB", "#E7B800", "#FC4E07")) + labs(x="Extent of ash tree disback", y = "Temerature range (°C)") + geom_jitter(shape=19, position=position_jitter(0.2)) + geom_point() + theme(legend.position = "none")
p

kruskal.test(Temp_range ~ AD, data = Ash)

    Kruskal-Wallis rank sum test

data:  Temp_range by AD
Kruskal-Wallis chi-squared = 15.28, df = 2, p-value = 0.0004809

pairwise.wilcox.test(Ash$Temp_range, Ash$AD, p.adjust.method = "BH")

cannot compute exact p-value with tiescannot compute exact p-value with tiescannot compute exact p-value with ties

    Pairwise comparisons using Wilcoxon rank sum test 

data:  Ash$Temp_range and Ash$AD 

       low    medium
medium 0.8371 -     
high   0.0021 0.0021

P value adjustment method: BH 

#significance between medium and high (0.0021)
#temp ~ LAI
ggplotRegression(lm(Canopy_cover ~ Temp_range, data = Ash))

ggplotRegression(lm(Canopy_cover ~ Min_temp, data = Ash))

ggplotRegression(lm(Canopy_cover ~ Max_temp, data = Ash))

#temp_range ~ Understorey (p=0.02)
ttest_range <- t.test(Temp_range ~ Understorey, data = Ash)
ttest_range

    Welch Two Sample t-test

data:  Temp_range by Understorey
t = 2.4451, df = 30.966, p-value = 0.02037
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.2673485 2.9569623
sample estimates:
mean in group - mean in group + 
       12.89286        11.28070 

#Max temp ~ understorey (p = 0.04)
ttest_max <- t.test(Max_temp ~ Understorey, data = Ash)
ttest_max

    Welch Two Sample t-test

data:  Max_temp by Understorey
t = 2.1134, df = 30.916, p-value = 0.04274
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.0467348 2.6337163
sample estimates:
mean in group - mean in group + 
       24.55952        23.21930 

#Min temp ~ understorey (p=0.0036)
ttest_min <- t.test(Min_temp ~ Understorey, data = Ash)
ttest_min

    Welch Two Sample t-test

data:  Min_temp by Understorey
t = -3.1499, df = 30.877, p-value = 0.003614
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.44803140 -0.09582825
sample estimates:
mean in group - mean in group + 
       11.66667        11.93860 

#GLM for temp range ~ AD * LAI
GLM_range_AD_LAI <- glm(Temp_range ~ AD * Canopy_cover, data = Ash)
summary(GLM_range_AD_LAI)

Call:
glm(formula = Temp_range ~ AD * Canopy_cover, data = Ash)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.0604  -0.9850  -0.4193   0.7151   2.6875  

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)           13.68377    2.41035   5.677 4.97e-06 ***
ADmedium              -5.66302    2.95658  -1.915   0.0661 .  
ADhigh                 2.65398    2.63664   1.007   0.3231    
Canopy_cover          -0.07059    0.05930  -1.190   0.2443    
ADmedium:Canopy_cover  0.14323    0.07399   1.936   0.0634 .  
ADhigh:Canopy_cover   -0.07683    0.07939  -0.968   0.3418    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 2.012371)

    Null deviance: 139.320  on 32  degrees of freedom
Residual deviance:  54.334  on 27  degrees of freedom
AIC: 124.11

Number of Fisher Scoring iterations: 2

GLM_range_AD_LAI_model <- stepAIC(GLM_range_AD_LAI, direction = "backward", Trace = FALSE)

Start:  AIC=124.11
Temp_range ~ AD * Canopy_cover

                  Df Deviance    AIC
<none>                 54.334 124.11
- AD:Canopy_cover  2   76.056 131.20

summary(GLM_range_AD_LAI_model)

Call:
glm(formula = Temp_range ~ AD * Canopy_cover, data = Ash)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.0604  -0.9850  -0.4193   0.7151   2.6875  

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)           13.68377    2.41035   5.677 4.97e-06 ***
ADmedium              -5.66302    2.95658  -1.915   0.0661 .  
ADhigh                 2.65398    2.63664   1.007   0.3231    
Canopy_cover          -0.07059    0.05930  -1.190   0.2443    
ADmedium:Canopy_cover  0.14323    0.07399   1.936   0.0634 .  
ADhigh:Canopy_cover   -0.07683    0.07939  -0.968   0.3418    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 2.012371)

    Null deviance: 139.320  on 32  degrees of freedom
Residual deviance:  54.334  on 27  degrees of freedom
AIC: 124.11

Number of Fisher Scoring iterations: 2

Anova(GLM_range_AD_LAI_model)

Analysis of Deviance Table (Type II tests)

Response: Temp_range
                LR Chisq Df Pr(>Chisq)   
AD                8.1816  2    0.01673 * 
Canopy_cover      1.1153  1    0.29094   
AD:Canopy_cover  10.7942  2    0.00453 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#GLM for temp range ~ AD * LAI * Understorey
GLM_range_AD_LAI_US <- glm(Temp_range ~ AD * Canopy_cover * Understorey, data = Ash)
summary(GLM_range_AD_LAI_US)

Call:
glm(formula = Temp_range ~ AD * Canopy_cover * Understorey, data = Ash)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.78877  -0.79380  -0.00246   0.42453   2.94694  

Coefficients:
                                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)                         15.34262    3.59668   4.266 0.000344 ***
ADmedium                           -33.94241   44.56550  -0.762 0.454750    
ADhigh                               0.45235    3.85771   0.117 0.907770    
Canopy_cover                        -0.08589    0.09303  -0.923 0.366371    
Understorey+                        -4.81944    4.81865  -1.000 0.328617    
ADmedium:Canopy_cover                0.93515    1.24276   0.752 0.460119    
ADhigh:Canopy_cover                 -0.03233    0.11597  -0.279 0.783118    
ADmedium:Understorey+               31.00502   44.71288   0.693 0.495642    
ADhigh:Understorey+                  6.01953    5.25033   1.147 0.264483    
Canopy_cover:Understorey+            0.07083    0.12016   0.589 0.561817    
ADmedium:Canopy_cover:Understorey+  -0.84392    1.24583  -0.677 0.505549    
ADhigh:Canopy_cover:Understorey+    -0.13526    0.15817  -0.855 0.402116    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 1.892805)

    Null deviance: 139.320  on 32  degrees of freedom
Residual deviance:  39.749  on 21  degrees of freedom
AIC: 125.79

Number of Fisher Scoring iterations: 2

GLM_range_AD_LAI_US_model <- stepAIC(GLM_range_AD_LAI_US, direction = "backward", Trace = FALSE)

Start:  AIC=125.79
Temp_range ~ AD * Canopy_cover * Understorey

                              Df Deviance    AIC
- AD:Canopy_cover:Understorey  2   41.852 123.49
<none>                             39.749 125.79

Step:  AIC=123.49
Temp_range ~ AD + Canopy_cover + Understorey + AD:Canopy_cover + 
    AD:Understorey + Canopy_cover:Understorey

                           Df Deviance    AIC
- AD:Understorey            2   43.421 120.71
- Canopy_cover:Understorey  1   41.885 121.52
<none>                          41.852 123.49
- AD:Canopy_cover           2   57.826 130.16

Step:  AIC=120.71
Temp_range ~ AD + Canopy_cover + Understorey + AD:Canopy_cover + 
    Canopy_cover:Understorey

                           Df Deviance    AIC
<none>                          43.421 120.71
- Canopy_cover:Understorey  1   47.858 121.92
- AD:Canopy_cover           2   69.476 132.22

summary(GLM_range_AD_LAI_US_model)

Call:
glm(formula = Temp_range ~ AD + Canopy_cover + Understorey + 
    AD:Canopy_cover + Canopy_cover:Understorey, data = Ash)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.87982  -0.89283   0.06472   0.43748   2.92255  

Coefficients:
                           Estimate Std. Error t value Pr(>|t|)    
(Intercept)               12.175030   2.435591   4.999 3.74e-05 ***
ADmedium                  -5.599137   2.789417  -2.007   0.0556 .  
ADhigh                     3.699850   2.492636   1.484   0.1502    
Canopy_cover              -0.007482   0.063814  -0.117   0.9076    
Understorey+               1.004402   1.319920   0.761   0.4538    
ADmedium:Canopy_cover      0.152287   0.069947   2.177   0.0391 *  
ADhigh:Canopy_cover       -0.108977   0.075069  -1.452   0.1590    
Canopy_cover:Understorey+ -0.068042   0.042572  -1.598   0.1225    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 1.736857)

    Null deviance: 139.320  on 32  degrees of freedom
Residual deviance:  43.421  on 25  degrees of freedom
AIC: 120.71

Number of Fisher Scoring iterations: 2

Anova(GLM_range_AD_LAI_US_model)

Analysis of Deviance Table (Type II tests)

Response: Temp_range
                         LR Chisq Df Pr(>Chisq)    
AD                         7.6994  2  0.0212856 *  
Canopy_cover               0.9391  1  0.3325155    
Understorey                3.7285  1  0.0534916 .  
AD:Canopy_cover           15.0008  2  0.0005529 ***
Canopy_cover:Understorey   2.5545  1  0.1099830    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

dropterm(GLM_range_AD_LAI_US_model,test = "Chisq")

Single term deletions

Model:
Temp_range ~ AD + Canopy_cover + Understorey + AD:Canopy_cover + 
    Canopy_cover:Understorey
                         Df Deviance    AIC scaled dev.   Pr(Chi)    
<none>                        43.421 120.71                          
AD:Canopy_cover           2   69.476 132.22     15.5108 0.0004284 ***
Canopy_cover:Understorey  1   47.858 121.92      3.2105 0.0731658 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

GLM_range_AD_LAI_US_model<- update(GLM_range_AD_LAI_US_model, . ~ .-Canopy_cover:Understorey)
Anova(GLM_range_AD_LAI_US_model)

Analysis of Deviance Table (Type II tests)

Response: Temp_range
                LR Chisq Df Pr(>Chisq)   
AD                7.3157  2    0.02579 * 
Canopy_cover      0.8861  1    0.34654   
Understorey       3.5182  1    0.06070 . 
AD:Canopy_cover  11.7995  2    0.00274 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#GLM for temp Max ~ AD * LAI
GLM_Max_AD_LAI <- glm(Max_temp ~ AD * Canopy_cover, data = Ash)
summary(GLM_Max_AD_LAI)

Call:
glm(formula = Max_temp ~ AD * Canopy_cover, data = Ash)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.8176  -0.8271  -0.3953   0.7564   2.9415  

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)           25.68355    2.33850  10.983 1.83e-11 ***
ADmedium              -6.09723    2.86845  -2.126   0.0428 *  
ADhigh                 2.21083    2.55805   0.864   0.3951    
Canopy_cover          -0.07337    0.05753  -1.275   0.2131    
ADmedium:Canopy_cover  0.15631    0.07179   2.177   0.0384 *  
ADhigh:Canopy_cover   -0.06698    0.07703  -0.870   0.3922    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 1.894189)

    Null deviance: 126.682  on 32  degrees of freedom
Residual deviance:  51.143  on 27  degrees of freedom
AIC: 122.11

Number of Fisher Scoring iterations: 2

GLM_Max_AD_LAI_model <- stepAIC(GLM_Max_AD_LAI, direction = "backward", Trace = FALSE)

Start:  AIC=122.11
Max_temp ~ AD * Canopy_cover

                  Df Deviance    AIC
<none>                 51.143 122.11
- AD:Canopy_cover  2   74.062 130.33

summary(GLM_Max_AD_LAI_model)

Call:
glm(formula = Max_temp ~ AD * Canopy_cover, data = Ash)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.8176  -0.8271  -0.3953   0.7564   2.9415  

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)           25.68355    2.33850  10.983 1.83e-11 ***
ADmedium              -6.09723    2.86845  -2.126   0.0428 *  
ADhigh                 2.21083    2.55805   0.864   0.3951    
Canopy_cover          -0.07337    0.05753  -1.275   0.2131    
ADmedium:Canopy_cover  0.15631    0.07179   2.177   0.0384 *  
ADhigh:Canopy_cover   -0.06698    0.07703  -0.870   0.3922    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 1.894189)

    Null deviance: 126.682  on 32  degrees of freedom
Residual deviance:  51.143  on 27  degrees of freedom
AIC: 122.11

Number of Fisher Scoring iterations: 2

Anova(GLM_Max_AD_LAI_model)

Analysis of Deviance Table (Type II tests)

Response: Max_temp
                LR Chisq Df Pr(>Chisq)   
AD                7.7053  2   0.021223 * 
Canopy_cover      0.7674  1   0.381034   
AD:Canopy_cover  12.0995  2   0.002358 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#GLM for temp Max ~ AD * LAI * Understorey
GLM_Max_AD_LAI_US <- glm(Max_temp ~ AD * Canopy_cover * Understorey, data = Ash)
summary(GLM_Max_AD_LAI_US)

Call:
glm(formula = Max_temp ~ AD * Canopy_cover * Understorey, data = Ash)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.9710  -0.5837  -0.1009   0.3097   3.1769  

Coefficients:
                                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)                         26.79143    3.60185   7.438 2.59e-07 ***
ADmedium                           -25.94961   44.62949  -0.581    0.567    
ADhigh                               0.44560    3.86325   0.115    0.909    
Canopy_cover                        -0.07795    0.09316  -0.837    0.412    
Understorey+                        -3.60592    4.82557  -0.747    0.463    
ADmedium:Canopy_cover                0.71490    1.24455   0.574    0.572    
ADhigh:Canopy_cover                 -0.03279    0.11613  -0.282    0.780    
ADmedium:Understorey+               21.94467   44.77708   0.490    0.629    
ADhigh:Understorey+                  5.07734    5.25787   0.966    0.345    
Canopy_cover:Understorey+            0.04689    0.12033   0.390    0.701    
ADmedium:Canopy_cover:Understorey+  -0.59748    1.24762  -0.479    0.637    
ADhigh:Canopy_cover:Understorey+    -0.11248    0.15840  -0.710    0.485    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 1.898244)

    Null deviance: 126.682  on 32  degrees of freedom
Residual deviance:  39.863  on 21  degrees of freedom
AIC: 125.89

Number of Fisher Scoring iterations: 2

GLM_Max_AD_LAI_US_model <- stepAIC(GLM_Max_AD_LAI_US, direction = "backward", Trace = FALSE)

Start:  AIC=125.89
Max_temp ~ AD * Canopy_cover * Understorey

                              Df Deviance    AIC
- AD:Canopy_cover:Understorey  2   41.168 122.95
<none>                             39.863 125.89

Step:  AIC=122.95
Max_temp ~ AD + Canopy_cover + Understorey + AD:Canopy_cover + 
    AD:Understorey + Canopy_cover:Understorey

                           Df Deviance    AIC
- AD:Understorey            2   42.417 119.93
- Canopy_cover:Understorey  1   41.294 121.05
<none>                          41.168 122.95
- AD:Canopy_cover           2   58.533 130.56

Step:  AIC=119.93
Max_temp ~ AD + Canopy_cover + Understorey + AD:Canopy_cover + 
    Canopy_cover:Understorey

                           Df Deviance    AIC
<none>                          42.417 119.93
- Canopy_cover:Understorey  1   47.405 121.60
- AD:Canopy_cover           2   70.065 132.50

summary(GLM_Max_AD_LAI_US_model)

Call:
glm(formula = Max_temp ~ AD + Canopy_cover + Understorey + AD:Canopy_cover + 
    Canopy_cover:Understorey, data = Ash)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.8336  -0.6947  -0.2494   0.4118   3.1440  

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)               24.07697    2.40724  10.002 3.19e-10 ***
ADmedium                  -6.22059    2.75695  -2.256   0.0330 *  
ADhigh                     3.20641    2.46362   1.301   0.2050    
Canopy_cover              -0.01028    0.06307  -0.163   0.8719    
Understorey+               1.35145    1.30456   1.036   0.3101    
ADmedium:Canopy_cover      0.16890    0.06913   2.443   0.0220 *  
ADhigh:Canopy_cover       -0.09738    0.07420  -1.312   0.2013    
Canopy_cover:Understorey+ -0.07215    0.04208  -1.715   0.0988 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 1.696661)

    Null deviance: 126.682  on 32  degrees of freedom
Residual deviance:  42.417  on 25  degrees of freedom
AIC: 119.93

Number of Fisher Scoring iterations: 2

Anova(GLM_Max_AD_LAI_US_model)

Analysis of Deviance Table (Type II tests)

Response: Max_temp
                         LR Chisq Df Pr(>Chisq)    
AD                         7.3700  2  0.0250973 *  
Canopy_cover               0.6320  1  0.4266055    
Understorey                2.2029  1  0.1377498    
AD:Canopy_cover           16.2957  2  0.0002894 ***
Canopy_cover:Understorey   2.9405  1  0.0863865 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

dropterm(GLM_Max_AD_LAI_US_model,test = "Chisq")

Single term deletions

Model:
Max_temp ~ AD + Canopy_cover + Understorey + AD:Canopy_cover + 
    Canopy_cover:Understorey
                         Df Deviance    AIC scaled dev.   Pr(Chi)    
<none>                        42.417 119.93                          
AD:Canopy_cover           2   70.065 132.50     16.5621 0.0002533 ***
Canopy_cover:Understorey  1   47.405 121.60      3.6696 0.0554138 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

GLM_Max_AD_LAI_US_model<- update(GLM_Max_AD_LAI_US_model, . ~ .-Canopy_cover:Understorey)
Anova(GLM_Max_AD_LAI_US_model)

Analysis of Deviance Table (Type II tests)

Response: Max_temp
                LR Chisq Df Pr(>Chisq)   
AD                6.8254  2   0.032952 * 
Canopy_cover      0.5882  1   0.443135   
Understorey       2.0499  1   0.152214   
AD:Canopy_cover  12.5201  2   0.001911 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

dropterm(GLM_Max_AD_LAI_US_model,test = "Chisq")

Single term deletions

Model:
Max_temp ~ AD + Canopy_cover + Understorey + AD:Canopy_cover
                Df Deviance    AIC scaled dev.  Pr(Chi)   
<none>               47.405 121.60                        
Understorey      1   51.143 122.11      2.5044 0.113532   
AD:Canopy_cover  2   70.233 130.57     12.9717 0.001525 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

GLM_Max_AD_LAI_US_model<- update(GLM_Max_AD_LAI_US_model, . ~ .-Understorey)
Anova(GLM_Max_AD_LAI_US_model)

Analysis of Deviance Table (Type II tests)

Response: Max_temp
                LR Chisq Df Pr(>Chisq)   
AD                7.7053  2   0.021223 * 
Canopy_cover      0.7674  1   0.381034   
AD:Canopy_cover  12.0995  2   0.002358 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#GLM for temp min ~ AD * LAI
GLM_Min_AD_LAI <- glm(Min_temp ~ AD * Canopy_cover, data = Ash)
summary(GLM_Min_AD_LAI)

Call:
glm(formula = Min_temp ~ AD * Canopy_cover, data = Ash)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-0.56263  -0.17650   0.04136   0.13546   0.40756  

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)           11.999780   0.463834  25.871   <2e-16 ***
ADmedium              -0.434208   0.568948  -0.763    0.452    
ADhigh                -0.443150   0.507381  -0.873    0.390    
Canopy_cover          -0.002782   0.011411  -0.244    0.809    
ADmedium:Canopy_cover  0.013078   0.014238   0.918    0.366    
ADhigh:Canopy_cover    0.009854   0.015278   0.645    0.524    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 0.07452019)

    Null deviance: 2.6077  on 32  degrees of freedom
Residual deviance: 2.0120  on 27  degrees of freedom
AIC: 15.337

Number of Fisher Scoring iterations: 2

GLM_Min_AD_LAI_model <- stepAIC(GLM_Min_AD_LAI, direction = "backward", Trace = FALSE)

Start:  AIC=15.34
Min_temp ~ AD * Canopy_cover

                  Df Deviance    AIC
- AD:Canopy_cover  2    2.076 12.369
<none>                  2.012 15.337

Step:  AIC=12.37
Min_temp ~ AD + Canopy_cover

               Df Deviance     AIC
- AD            2   2.1527  9.5664
- Canopy_cover  1   2.1615 11.7018
<none>              2.0760 12.3692

Step:  AIC=9.57
Min_temp ~ Canopy_cover

               Df Deviance     AIC
<none>              2.1527  9.5664
- Canopy_cover  1   2.6077 13.8952

summary(GLM_Min_AD_LAI_model)

Call:
glm(formula = Min_temp ~ Canopy_cover, data = Ash)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-0.50704  -0.15187   0.02257   0.15022   0.54313  

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  11.547142   0.117198   98.53   <2e-16 ***
Canopy_cover  0.009138   0.003569    2.56   0.0156 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 0.06944061)

    Null deviance: 2.6077  on 32  degrees of freedom
Residual deviance: 2.1527  on 31  degrees of freedom
AIC: 9.5664

Number of Fisher Scoring iterations: 2

Anova(GLM_Min_AD_LAI_model)

Analysis of Deviance Table (Type II tests)

Response: Min_temp
             LR Chisq Df Pr(>Chisq)  
Canopy_cover   6.5536  1    0.01047 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#GLM for temp Min ~ AD * LAI * Understorey
GLM_Min_AD_LAI_US <- glm(Min_temp ~ AD * Canopy_cover * Understorey, data = Ash)
summary(GLM_Min_AD_LAI_US)

Call:
glm(formula = Min_temp ~ AD * Canopy_cover * Understorey, data = Ash)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-0.58851  -0.11486   0.04274   0.16721   0.37070  

Coefficients:
                                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)                        11.4488113  0.6940087  16.497  1.7e-13 ***
ADmedium                            7.9928023  8.5992744   0.929    0.363    
ADhigh                             -0.0067449  0.7443768  -0.009    0.993    
Canopy_cover                        0.0079370  0.0179510   0.442    0.663    
Understorey+                        1.2135219  0.9297979   1.305    0.206    
ADmedium:Canopy_cover              -0.2202512  0.2398013  -0.918    0.369    
ADhigh:Canopy_cover                -0.0004561  0.0223764  -0.020    0.984    
ADmedium:Understorey+              -9.0603475  8.6277122  -1.050    0.306    
ADhigh:Understorey+                -0.9421920  1.0130936  -0.930    0.363    
Canopy_cover:Understorey+          -0.0239408  0.0231851  -1.033    0.314    
ADmedium:Canopy_cover:Understorey+  0.2464467  0.2403934   1.025    0.317    
ADhigh:Canopy_cover:Understorey+    0.0227779  0.0305198   0.746    0.464    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 0.07047446)

    Null deviance: 2.6077  on 32  degrees of freedom
Residual deviance: 1.4800  on 21  degrees of freedom
AIC: 17.202

Number of Fisher Scoring iterations: 2

GLM_Min_AD_LAI_US_model <- stepAIC(GLM_Min_AD_LAI_US, direction = "backward", Trace = FALSE)

Start:  AIC=17.2
Min_temp ~ AD * Canopy_cover * Understorey

                              Df Deviance    AIC
- AD:Canopy_cover:Understorey  2   1.5859 15.484
<none>                             1.4800 17.202

Step:  AIC=15.48
Min_temp ~ AD + Canopy_cover + Understorey + AD:Canopy_cover + 
    AD:Understorey + Canopy_cover:Understorey

                           Df Deviance    AIC
- AD:Understorey            2   1.6219 12.225
- AD:Canopy_cover           2   1.7112 13.993
- Canopy_cover:Understorey  1   1.6163 14.110
<none>                          1.5859 15.484

Step:  AIC=12.22
Min_temp ~ AD + Canopy_cover + Understorey + AD:Canopy_cover + 
    Canopy_cover:Understorey

                           Df Deviance    AIC
- AD:Canopy_cover           2   1.7212 10.184
- Canopy_cover:Understorey  1   1.6381 10.553
<none>                          1.6219 12.225

Step:  AIC=10.18
Min_temp ~ AD + Canopy_cover + Understorey + Canopy_cover:Understorey

                           Df Deviance     AIC
- AD                        2   1.7392  6.5278
- Canopy_cover:Understorey  1   1.7295  8.3433
<none>                          1.7212 10.1841

Step:  AIC=6.53
Min_temp ~ Canopy_cover + Understorey + Canopy_cover:Understorey

                           Df Deviance    AIC
- Canopy_cover:Understorey  1   1.7517 4.7650
<none>                          1.7392 6.5278

Step:  AIC=4.76
Min_temp ~ Canopy_cover + Understorey

               Df Deviance    AIC
<none>              1.7517 4.7650
- Canopy_cover  1   2.0117 7.3315
- Understorey   1   2.1527 9.5664

summary(GLM_Min_AD_LAI_US_model)

Call:
glm(formula = Min_temp ~ Canopy_cover + Understorey, data = Ash)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-0.60047  -0.13191  -0.01464   0.17141   0.43851  

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  11.476757   0.110776  103.60   <2e-16 ***
Canopy_cover  0.007099   0.003364    2.11   0.0433 *  
Understorey+  0.229240   0.087483    2.62   0.0137 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 0.05839075)

    Null deviance: 2.6077  on 32  degrees of freedom
Residual deviance: 1.7517  on 30  degrees of freedom
AIC: 4.765

Number of Fisher Scoring iterations: 2

Anova(GLM_Min_AD_LAI_US_model)

Analysis of Deviance Table (Type II tests)

Response: Min_temp
             LR Chisq Df Pr(>Chisq)   
Canopy_cover   4.4523  1   0.034854 * 
Understorey    6.8664  1   0.008783 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

dropterm(GLM_Min_AD_LAI_US_model,test = "Chisq")

Single term deletions

Model:
Min_temp ~ Canopy_cover + Understorey
             Df Deviance    AIC scaled dev.  Pr(Chi)   
<none>            1.7517 4.7650                        
Canopy_cover  1   2.0117 7.3315      4.5665 0.032603 * 
Understorey   1   2.1527 9.5664      6.8014 0.009108 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Factor 3: Soil moisture ~ AD

p <- ggqqplot(Ash$Soil_moisture_ave)
p

shapiro.test(Ash$Soil_moisture_ave) #the data is normally distributed

    Shapiro-Wilk normality test

data:  Ash$Soil_moisture_ave
W = 0.94762, p-value = 0.1133

p <- ggboxplot(Ash, x="AD", y= "Soil_moisture_ave",color = "AD", palette = c("#00AFBB", "#E7B800", "#FC4E07")) + labs(x="Extent of ash tree disback", y = "Soil moisture(m^3/m^3)") + geom_jitter(shape=19, position=position_jitter(0.2)) + geom_point() + theme(legend.position = "none")
p

SM_AD <- aov(Soil_moisture_ave ~ AD,data = Ash)
summary(SM_AD)

            Df  Sum Sq  Mean Sq F value Pr(>F)  
AD           2 0.01015 0.005074   3.288 0.0511 .
Residuals   30 0.04629 0.001543                 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

TukeyHSD(SM_AD)#significance between low and high(p=0.042)

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = Soil_moisture_ave ~ AD, data = Ash)

$AD
                  diff          lwr        upr     p adj
medium-low  0.03041111 -0.014082207 0.07490443 0.2274018
high-low    0.04275397  0.001380884 0.08412705 0.0416947
high-medium 0.01234286 -0.027751278 0.05243699 0.7306348

#soil moisture vs LAI
ggplotRegression(lm(Canopy_cover ~ Soil_moisture_ave, data = Ash))

ggplotRegression(lm(Temp_range ~ Soil_moisture_ave, data = Ash))

#Soil moisture ~ Understorey (no significant difference)
ttest_soil <- t.test(Soil_moisture_ave ~ Understorey, data = Ash)
ttest_soil

    Welch Two Sample t-test

data:  Soil_moisture_ave by Understorey
t = 0.034764, df = 18.24, p-value = 0.9726
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.03392985  0.03507271
sample estimates:
mean in group - mean in group + 
      0.1635714       0.1630000 

#GLM for soil ~ AD * LAI * temp
GLM_soil_AD_LAI_T <- glm(Soil_moisture_ave ~ AD * Canopy_cover * Temp_range, data = Ash)
summary(GLM_soil_AD_LAI_T)

Call:
glm(formula = Soil_moisture_ave ~ AD * Canopy_cover * Temp_range, 
    data = Ash)

Deviance Residuals: 
      Min         1Q     Median         3Q        Max  
-0.081414  -0.015334  -0.000157   0.009398   0.079531  

Coefficients:
                                  Estimate Std. Error t value Pr(>|t|)  
(Intercept)                      -0.921634   0.800870  -1.151   0.2628  
ADmedium                          1.597247   0.866373   1.844   0.0794 .
ADhigh                            1.257761   0.855518   1.470   0.1563  
Canopy_cover                      0.029502   0.021059   1.401   0.1758  
Temp_range                        0.099348   0.074217   1.339   0.1950  
ADmedium:Canopy_cover            -0.042537   0.022391  -1.900   0.0713 .
ADhigh:Canopy_cover              -0.031984   0.024499  -1.306   0.2058  
ADmedium:Temp_range              -0.153826   0.081514  -1.887   0.0730 .
ADhigh:Temp_range                -0.106541   0.077253  -1.379   0.1824  
Canopy_cover:Temp_range          -0.002795   0.001983  -1.409   0.1735  
ADmedium:Canopy_cover:Temp_range  0.004172   0.002123   1.965   0.0628 .
ADhigh:Canopy_cover:Temp_range    0.002742   0.002197   1.248   0.2257  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 0.001540647)

    Null deviance: 0.056436  on 32  degrees of freedom
Residual deviance: 0.032354  on 21  degrees of freedom
AIC: -108.96

Number of Fisher Scoring iterations: 2

GLM_soil_AD_LAI_T_model <- stepAIC(GLM_soil_AD_LAI_T, direction = "backward", Trace = FALSE)

Start:  AIC=-108.96
Soil_moisture_ave ~ AD * Canopy_cover * Temp_range

                             Df Deviance     AIC
<none>                          0.032354 -108.96
- AD:Canopy_cover:Temp_range  2 0.039195 -106.63

summary(GLM_soil_AD_LAI_T_model)

Call:
glm(formula = Soil_moisture_ave ~ AD * Canopy_cover * Temp_range, 
    data = Ash)

Deviance Residuals: 
      Min         1Q     Median         3Q        Max  
-0.081414  -0.015334  -0.000157   0.009398   0.079531  

Coefficients:
                                  Estimate Std. Error t value Pr(>|t|)  
(Intercept)                      -0.921634   0.800870  -1.151   0.2628  
ADmedium                          1.597247   0.866373   1.844   0.0794 .
ADhigh                            1.257761   0.855518   1.470   0.1563  
Canopy_cover                      0.029502   0.021059   1.401   0.1758  
Temp_range                        0.099348   0.074217   1.339   0.1950  
ADmedium:Canopy_cover            -0.042537   0.022391  -1.900   0.0713 .
ADhigh:Canopy_cover              -0.031984   0.024499  -1.306   0.2058  
ADmedium:Temp_range              -0.153826   0.081514  -1.887   0.0730 .
ADhigh:Temp_range                -0.106541   0.077253  -1.379   0.1824  
Canopy_cover:Temp_range          -0.002795   0.001983  -1.409   0.1735  
ADmedium:Canopy_cover:Temp_range  0.004172   0.002123   1.965   0.0628 .
ADhigh:Canopy_cover:Temp_range    0.002742   0.002197   1.248   0.2257  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 0.001540647)

    Null deviance: 0.056436  on 32  degrees of freedom
Residual deviance: 0.032354  on 21  degrees of freedom
AIC: -108.96

Number of Fisher Scoring iterations: 2

Anova(GLM_soil_AD_LAI_T_model)

Analysis of Deviance Table (Type II tests)

Response: Soil_moisture_ave
                           LR Chisq Df Pr(>Chisq)
AD                           3.3845  2     0.1841
Canopy_cover                 0.3254  1     0.5684
Temp_range                   0.2573  1     0.6120
AD:Canopy_cover              3.4023  2     0.1825
AD:Temp_range                0.1561  2     0.9249
Canopy_cover:Temp_range      0.8539  1     0.3555
AD:Canopy_cover:Temp_range   4.4405  2     0.1086

#GLM for soil ~ AD * LAI * Temp * Understorey
GLM_soil_AD_LAI_US_T <- glm(Soil_moisture_ave ~ AD * Canopy_cover * Understorey * Temp_range, data = Ash)
summary(GLM_soil_AD_LAI_US_T)

Call:
glm(formula = Soil_moisture_ave ~ AD * Canopy_cover * Understorey * 
    Temp_range, data = Ash)

Deviance Residuals: 
      Min         1Q     Median         3Q        Max  
-0.091963  -0.005916   0.000000   0.007173   0.067235  

Coefficients: (3 not defined because of singularities)
                                               Estimate Std. Error t value Pr(>|t|)
(Intercept)                                   -1.410475   3.132171  -0.450    0.661
ADmedium                                      -4.641804   9.130635  -0.508    0.620
ADhigh                                         2.045804   3.207693   0.638    0.536
Canopy_cover                                   0.040000   0.075404   0.530    0.605
Understorey+                                  -4.247619   4.497782  -0.944    0.364
Temp_range                                     0.138347   0.261234   0.530    0.606
ADmedium:Canopy_cover                          0.254839   0.377279   0.675    0.512
ADhigh:Canopy_cover                           -0.054820   0.081101  -0.676    0.512
ADmedium:Understorey+                         10.975472  12.160898   0.903    0.385
ADhigh:Understorey+                            3.769302   4.460876   0.845    0.415
Canopy_cover:Understorey+                      0.017857   0.029918   0.597    0.562
ADmedium:Temp_range                           -0.608677   0.441100  -1.380    0.193
ADhigh:Temp_range                             -0.166987   0.266042  -0.628    0.542
Canopy_cover:Temp_range                       -0.003638   0.006381  -0.570    0.579
Understorey+:Temp_range                        0.415972   0.414460   1.004    0.335
ADmedium:Canopy_cover:Understorey+            -0.325623   0.359603  -0.906    0.383
ADhigh:Canopy_cover:Understorey+               0.001494   0.017834   0.084    0.935
ADmedium:Canopy_cover:Temp_range               0.006459   0.006999   0.923    0.374
ADhigh:Canopy_cover:Temp_range                 0.004513   0.006763   0.667    0.517
ADmedium:Understorey+:Temp_range                     NA         NA      NA       NA
ADhigh:Understorey+:Temp_range                -0.382201   0.409328  -0.934    0.369
Canopy_cover:Understorey+:Temp_range          -0.001455   0.002648  -0.550    0.593
ADmedium:Canopy_cover:Understorey+:Temp_range        NA         NA      NA       NA
ADhigh:Canopy_cover:Understorey+:Temp_range          NA         NA      NA       NA

(Dispersion parameter for gaussian family taken to be 0.002317415)

    Null deviance: 0.056436  on 32  degrees of freedom
Residual deviance: 0.027809  on 12  degrees of freedom
AIC: -95.954

Number of Fisher Scoring iterations: 2

GLM_soil_AD_LAI_US_T_model <- stepAIC(GLM_soil_AD_LAI_US_T, direction = "backward", Trace = FALSE)

Start:  AIC=-95.95
Soil_moisture_ave ~ AD * Canopy_cover * Understorey * Temp_range


Step:  AIC=-95.95
Soil_moisture_ave ~ AD + Canopy_cover + Understorey + Temp_range + 
    AD:Canopy_cover + AD:Understorey + Canopy_cover:Understorey + 
    AD:Temp_range + Canopy_cover:Temp_range + Understorey:Temp_range + 
    AD:Canopy_cover:Understorey + AD:Canopy_cover:Temp_range + 
    AD:Understorey:Temp_range + Canopy_cover:Understorey:Temp_range

                                      Df Deviance     AIC
- AD:Canopy_cover:Understorey          1 0.027825 -97.935
- Canopy_cover:Understorey:Temp_range  1 0.028509 -97.134
<none>                                   0.027809 -95.954
- AD:Canopy_cover:Temp_range           2 0.031546 -95.793
- AD:Understorey:Temp_range            1 0.029829 -95.639

Step:  AIC=-97.93
Soil_moisture_ave ~ AD + Canopy_cover + Understorey + Temp_range + 
    AD:Canopy_cover + AD:Understorey + Canopy_cover:Understorey + 
    AD:Temp_range + Canopy_cover:Temp_range + Understorey:Temp_range + 
    AD:Canopy_cover:Temp_range + AD:Understorey:Temp_range + 
    Canopy_cover:Understorey:Temp_range

                                      Df Deviance     AIC
- Canopy_cover:Understorey:Temp_range  1 0.028608 -99.019
- AD:Understorey:Temp_range            2 0.030794 -98.589
<none>                                   0.027825 -97.935
- AD:Canopy_cover:Temp_range           2 0.034539 -94.802

Step:  AIC=-99.02
Soil_moisture_ave ~ AD + Canopy_cover + Understorey + Temp_range + 
    AD:Canopy_cover + AD:Understorey + Canopy_cover:Understorey + 
    AD:Temp_range + Canopy_cover:Temp_range + Understorey:Temp_range + 
    AD:Canopy_cover:Temp_range + AD:Understorey:Temp_range

                             Df Deviance      AIC
- Canopy_cover:Understorey    1 0.028651 -100.969
- AD:Understorey:Temp_range   2 0.031078 -100.286
<none>                          0.028608  -99.019
- AD:Canopy_cover:Temp_range  2 0.034826  -96.528

Step:  AIC=-100.97
Soil_moisture_ave ~ AD + Canopy_cover + Understorey + Temp_range + 
    AD:Canopy_cover + AD:Understorey + AD:Temp_range + Canopy_cover:Temp_range + 
    Understorey:Temp_range + AD:Canopy_cover:Temp_range + AD:Understorey:Temp_range

                             Df Deviance     AIC
- AD:Understorey:Temp_range   2 0.031161 -102.20
<none>                          0.028651 -100.97
- AD:Canopy_cover:Temp_range  2 0.036844  -96.67

Step:  AIC=-102.2
Soil_moisture_ave ~ AD + Canopy_cover + Understorey + Temp_range + 
    AD:Canopy_cover + AD:Understorey + AD:Temp_range + Canopy_cover:Temp_range + 
    Understorey:Temp_range + AD:Canopy_cover:Temp_range

                             Df Deviance     AIC
- AD:Understorey              2 0.031721 -105.61
- Understorey:Temp_range      1 0.031178 -104.18
<none>                          0.031161 -102.20
- AD:Canopy_cover:Temp_range  2 0.037025 -100.51

Step:  AIC=-105.61
Soil_moisture_ave ~ AD + Canopy_cover + Understorey + Temp_range + 
    AD:Canopy_cover + AD:Temp_range + Canopy_cover:Temp_range + 
    Understorey:Temp_range + AD:Canopy_cover:Temp_range

                             Df Deviance     AIC
- Understorey:Temp_range      1 0.031944 -107.38
<none>                          0.031721 -105.61
- AD:Canopy_cover:Temp_range  2 0.038694 -103.05

Step:  AIC=-107.38
Soil_moisture_ave ~ AD + Canopy_cover + Understorey + Temp_range + 
    AD:Canopy_cover + AD:Temp_range + Canopy_cover:Temp_range + 
    AD:Canopy_cover:Temp_range

                             Df Deviance     AIC
- Understorey                 1 0.032354 -108.96
<none>                          0.031944 -107.38
- AD:Canopy_cover:Temp_range  2 0.039192 -104.63

Step:  AIC=-108.96
Soil_moisture_ave ~ AD + Canopy_cover + Temp_range + AD:Canopy_cover + 
    AD:Temp_range + Canopy_cover:Temp_range + AD:Canopy_cover:Temp_range

                             Df Deviance     AIC
<none>                          0.032354 -108.96
- AD:Canopy_cover:Temp_range  2 0.039195 -106.63

summary(GLM_soil_AD_LAI_US_T_model)

Call:
glm(formula = Soil_moisture_ave ~ AD + Canopy_cover + Temp_range + 
    AD:Canopy_cover + AD:Temp_range + Canopy_cover:Temp_range + 
    AD:Canopy_cover:Temp_range, data = Ash)

Deviance Residuals: 
      Min         1Q     Median         3Q        Max  
-0.081414  -0.015334  -0.000157   0.009398   0.079531  

Coefficients:
                                  Estimate Std. Error t value Pr(>|t|)  
(Intercept)                      -0.921634   0.800870  -1.151   0.2628  
ADmedium                          1.597247   0.866373   1.844   0.0794 .
ADhigh                            1.257761   0.855518   1.470   0.1563  
Canopy_cover                      0.029502   0.021059   1.401   0.1758  
Temp_range                        0.099348   0.074217   1.339   0.1950  
ADmedium:Canopy_cover            -0.042537   0.022391  -1.900   0.0713 .
ADhigh:Canopy_cover              -0.031984   0.024499  -1.306   0.2058  
ADmedium:Temp_range              -0.153826   0.081514  -1.887   0.0730 .
ADhigh:Temp_range                -0.106541   0.077253  -1.379   0.1824  
Canopy_cover:Temp_range          -0.002795   0.001983  -1.409   0.1735  
ADmedium:Canopy_cover:Temp_range  0.004172   0.002123   1.965   0.0628 .
ADhigh:Canopy_cover:Temp_range    0.002742   0.002197   1.248   0.2257  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 0.001540647)

    Null deviance: 0.056436  on 32  degrees of freedom
Residual deviance: 0.032354  on 21  degrees of freedom
AIC: -108.96

Number of Fisher Scoring iterations: 2

Anova(GLM_soil_AD_LAI_US_T_model)

Analysis of Deviance Table (Type II tests)

Response: Soil_moisture_ave
                           LR Chisq Df Pr(>Chisq)
AD                           3.3845  2     0.1841
Canopy_cover                 0.3254  1     0.5684
Temp_range                   0.2573  1     0.6120
AD:Canopy_cover              3.4023  2     0.1825
AD:Temp_range                0.1561  2     0.9249
Canopy_cover:Temp_range      0.8539  1     0.3555
AD:Canopy_cover:Temp_range   4.4405  2     0.1086

dropterm(GLM_soil_AD_LAI_US_T_model,test = "Chisq")

Single term deletions

Model:
Soil_moisture_ave ~ AD + Canopy_cover + Temp_range + AD:Canopy_cover + 
    AD:Temp_range + Canopy_cover:Temp_range + AD:Canopy_cover:Temp_range
                           Df Deviance     AIC scaled dev. Pr(Chi)  
<none>                        0.032354 -108.96                      
AD:Canopy_cover:Temp_range  2 0.039195 -106.63        6.33 0.04221 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Factor 4: herb layer (NO MORE INTERACTION FROM NOW ON, MODEL OVERFIT)

Anova(Ash_glm)

Analysis of Deviance Table (Type II tests)

Response: cbind(Ash_percent, (100 - Ash_percent))
                  LR Chisq Df Pr(>Chisq)
AD                 1.62824  2     0.4430
Tree_dens          0.18887  1     0.6639
Canopy_cover       1.14095  1     0.2855
Temp_range         0.37471  1     0.5404
Soil_moisture_ave  0.10502  1     0.7459

Factor 5: invertebrate species