Stat3: Demo

Veröffentlichungsdatum

6. November 2023

Download dieses Demoscript via “</>Code” (oben rechts)
Datensatz ipomopsis.csv
Datensatz loyn.csv

ANCOVA

Experiment zur Fruchtproduktion (“Fruit”) von Ipomopsis sp. in Abhängigkeit von der Beweidung (“Grazing” mit 2 Levels: “Grazed”, “Ungrazed”) und korrigiert für die Pflanzengrösse vor der Beweidung (hier ausgedrückt als Durchmesser an der Spitze des Wurzelstock: “Root”)

# Daten einlesen und anschauen
library("readr")

compensation <- read_delim("datasets/stat1-4/ipomopsis.csv", ",",  col_types = cols("Grazing" = "f"))
head(compensation)

# A tibble: 6 × 4
   ...1  Root Fruit Grazing 
  <dbl> <dbl> <dbl> <fct>   
1     1  6.22  59.8 Ungrazed
2     2  6.49  61.0 Ungrazed
3     3  4.92  14.7 Ungrazed
4     4  5.13  19.3 Ungrazed
5     5  5.42  34.2 Ungrazed
6     6  5.36  35.5 Ungrazed

summary(compensation)

      ...1            Root            Fruit            Grazing  
 Min.   : 1.00   Min.   : 4.426   Min.   : 14.73   Ungrazed:20  
 1st Qu.:10.75   1st Qu.: 6.083   1st Qu.: 41.15   Grazed  :20  
 Median :20.50   Median : 7.123   Median : 60.88                
 Mean   :20.50   Mean   : 7.181   Mean   : 59.41                
 3rd Qu.:30.25   3rd Qu.: 8.510   3rd Qu.: 76.19                
 Max.   :40.00   Max.   :10.253   Max.   :116.05

# Pflanzengrösse ("Root") vs. Fruchtproduktion ("Fruit")
plot(Fruit ~ Root, data = compensation)

-> Je grösser die Pflanze, desto grösser ihre Fruchtproduktion.

# Beweidung ("Grazing") vs. Fruchtroduktion ("Fruit)
boxplot(Fruit ~ Grazing, data = compensation)

-> In der beweideten Gruppe scheint die Fruchtproduktion grösser. Liegt dies an der Beweidung oder an unterschiedlichen Pflanzengrössen zwischen den Gruppen?

# Plotten der vollständigen Daten/Information
library("ggplot2")
ggplot(compensation, aes(Root, Fruit, color = Grazing)) +
  geom_point() +
  theme_classic()

-> Die grössere Fruchtproduktion innerhalb der beweideten Gruppe scheint also ein Resultat von unterschiedlichen Pflanzengrössen zwischen den Gruppen zu sein und nicht an der Beweidung zu liegen.

# Lineare Modelle definieren und anschauen

aoc.1 <- lm(Fruit ~ Root * Grazing, data = compensation) # Volles Modell mit Interaktion
summary.aov(aoc.1)

             Df Sum Sq Mean Sq F value   Pr(>F)    
Root          1  16795   16795 359.968  < 2e-16 ***
Grazing       1   5264    5264 112.832 1.21e-12 ***
Root:Grazing  1      5       5   0.103     0.75    
Residuals    36   1680      47                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

aoc.2 <- lm(Fruit ~ Grazing + Root, data = compensation) # Finales Modell ohne die (nicht signifikante) Interaktion
summary.aov(aoc.2) # ANOVA-Tabelle

            Df Sum Sq Mean Sq F value  Pr(>F)    
Grazing      1   2910    2910   63.93 1.4e-09 ***
Root         1  19149   19149  420.62 < 2e-16 ***
Residuals   37   1684      46                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary(aoc.2) # Parameter-Tabelle


Call:
lm(formula = Fruit ~ Grazing + Root, data = compensation)

Residuals:
     Min       1Q   Median       3Q      Max 
-17.1920  -2.8224   0.3223   3.9144  17.3290 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)    -91.726      7.115  -12.89 2.96e-15 ***
GrazingGrazed  -36.103      3.357  -10.75 6.11e-13 ***
Root            23.560      1.149   20.51  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.747 on 37 degrees of freedom
Multiple R-squared:  0.9291,    Adjusted R-squared:  0.9252 
F-statistic: 242.3 on 2 and 37 DF,  p-value: < 2.2e-16

# Residualplots anschauen
par(mfrow = c(2, 2))
plot(aoc.2)

-> Das ANCOVA-Modell widerspiegelt die Zusammenhänge wie sie aufgrund der grafisch dargestellten Daten zu vermuten sind gut. Die Residual-Plots zeigen 3 Ausreisser (Beobachtungen 27, 34 und 37), welche “aus der Reihe tanzen”.

Polynomische Regression

# Daten generieren und Modelle rechnen
pred <- c(20, 19, 25, 10, 8, 15, 13, 18, 11, 14, 25, 39, 38, 28, 24) # "pred" sei unsere unabhängige Variable
resp <- c(12, 15, 10, 7, 2, 10, 12, 11, 13, 10, 9, 2, 4, 7, 13) # "resp" sei unsere abhängige Variable

plot(pred, resp) # So sehen die Daten aus

# Modelle definieren
lm.1 <- lm(resp ~ pred) # Einfaches lineares Modell
lm.quad <- lm(resp ~ pred + I(pred^2)) # lineares Modell mit quadratischem Term

summary(lm.1) # Modell anschauen


Call:
lm(formula = resp ~ pred)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.0549 -1.7015  0.5654  2.0617  5.6406 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  12.2879     2.4472   5.021 0.000234 ***
pred         -0.1541     0.1092  -1.412 0.181538    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.863 on 13 degrees of freedom
Multiple R-squared:  0.1329,    Adjusted R-squared:  0.06622 
F-statistic: 1.993 on 1 and 13 DF,  p-value: 0.1815

-> kein signifikanter Zusammenhang im einfachen linearen Modell und entsprechend kleines Bestimmtheitsmass (adj. R² = 0.07)

summary(lm.quad) # lineares Modell mit quadratischem Term anschauen


Call:
lm(formula = resp ~ pred + I(pred^2))

Residuals:
    Min      1Q  Median      3Q     Max 
-4.3866 -1.1018 -0.2027  1.3831  4.4211 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)   
(Intercept) -2.239308   3.811746  -0.587  0.56777   
pred         1.330933   0.360105   3.696  0.00306 **
I(pred^2)   -0.031587   0.007504  -4.209  0.00121 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.555 on 12 degrees of freedom
Multiple R-squared:  0.6499,    Adjusted R-squared:  0.5915 
F-statistic: 11.14 on 2 and 12 DF,  p-value: 0.001842

-> signifikanter Zusammenhang und viel besseres Bestimmtheitsmass (adj. R² = 0.60)

# Modelle plotten

par(mfrow = c(1, 2))

# 1. lineares Modell
plot(resp ~ pred, main = "Lineares Modell")
abline(lm.1, col = "blue")

# 2. quadratisches Modell
plot(resp ~ pred, main = "Quadratisches  Modell")
xv <- seq(0, 40, 0.1) # Input für Modellvoraussage via predict ()
yv2 <- predict(lm.quad, list(pred = xv))
lines(xv, yv2, col = "red")

# Residualplots
par(mfrow = c(2, 2))
plot(lm.1, main = "Lineares Modell")

plot(lm.quad, main = "Quadratisches  Modell")

-> Die Plots sehen beim Modell mit quadratischem Term besser aus

Simulation Overfitting

# Beispieldaten mit 6 Datenpunkten
test <- data.frame("x" = c(1, 2, 3, 4, 5, 6), "y" = c(34, 21, 70, 47, 23, 45))

par(mfrow = c(1, 1))
plot(y ~ x, data = test)

# Zunehmend komplizierte Modelle (je komplizierter desto overfitteter) definieren
lm.0 <- lm(y ~ 1, data = test) # Modell 
lm.1 <- lm(y ~ x, data = test)
lm.2 <- lm(y ~ x + I(x^2), data = test)
lm.3 <- lm(y ~ x + I(x^2) + I(x^3), data = test)
lm.4 <- lm(y ~ x + I(x^2) + I(x^3) + I(x^4), data = test)
lm.5 <- lm(y ~ x + I(x^2) + I(x^3) + I(x^4) + I(x^5), data = test)

# Summaries rechnen
smy.0 <- summary(lm.0)
smy.1 <- summary(lm.1)
smy.2 <- summary(lm.2)
smy.3 <- summary(lm.3)
smy.4 <- summary(lm.4)
smy.5 <- summary(lm.5)

# R2 vergleichen

smy.0$r.squared

[1] 0

smy.1$r.squared

[1] 0.01242685

smy.2$r.squared

[1] 0.1105981

smy.3$r.squared

[1] 0.1697982

smy.4$r.squared

[1] 0.874639

smy.5$r.squared

[1] 1

-> R2 wird immer grösser, d.h. die Modelle werden immer besser. ;-)

# Modelle plotten
xv <- seq(from = 0, to = 10, by = 0.1)
plot(y ~ x, cex = 2, col = "black", lwd = 3, data = test)

abline(h = lm.0$coefficients, col = "darkgray", lwd = 3 )
text(x = 1.5, y = 70, "lm.0", col = "darkgray")
yv <- predict(lm.1, list(x = xv))
lines(xv, yv, col = "red", lwd = 3)
text(x = 1.5, y = 67, "lm.1", col = "red")
yv <- predict(lm.2, list(x = xv))
lines(xv, yv, col = "blue", lwd = 3)
text(x = 1.5, y = 64, "lm.2", col = "blue")
yv <- predict(lm.3, list(x = xv))
lines(xv, yv, col = "green", lwd = 3)
text(x = 1.5, y = 61, "lm.3", col = "green")
yv <- predict(lm.4, list(x = xv))
lines(xv, yv, col = "orange", lwd = 3)
text(x = 1.5, y = 58, "lm.4", col = "orange")
yv <- predict(lm.5, list(x = xv))
lines(xv, yv, col = "violet", lwd = 3)
text(x = 1.5, y = 55, "lm.5", col = "violet")

-> Auch der optische Fit wird immer besser. Wir bestreiben jedoch Overfitting und Overfittig ist nicht gut: Denn, macht es Sinn, 6 Datenpunkte mit einem Modell mit 6 geschätzen Parametern zu fitten?? Die zunehmend komplizierten Modelle beschreiben lediglich die vorhandenen Datenpunkte, während sich die Voraussagekraft für weitere aus der Realität stammenden Daten zunehmend verschlechtert.

Multiple lineare Regression (basierend auf Logan (2010), Beispiel 9A)

# Daten laden und anschauen
library("readr")

loyn <- read_delim("datasets/stat1-4/loyn.csv", ",")
summary(loyn)

      ...1           ABUND            AREA            YR.ISOL    
 Min.   : 1.00   Min.   : 1.50   Min.   :   0.10   Min.   :1890  
 1st Qu.:14.75   1st Qu.:12.40   1st Qu.:   2.00   1st Qu.:1928  
 Median :28.50   Median :21.05   Median :   7.50   Median :1962  
 Mean   :28.50   Mean   :19.51   Mean   :  69.27   Mean   :1950  
 3rd Qu.:42.25   3rd Qu.:28.30   3rd Qu.:  29.75   3rd Qu.:1966  
 Max.   :56.00   Max.   :39.60   Max.   :1771.00   Max.   :1976  
      DIST            LDIST            GRAZE            ALT       
 Min.   :  26.0   Min.   :  26.0   Min.   :1.000   Min.   : 60.0  
 1st Qu.:  93.0   1st Qu.: 158.2   1st Qu.:2.000   1st Qu.:120.0  
 Median : 234.0   Median : 338.5   Median :3.000   Median :140.0  
 Mean   : 240.4   Mean   : 733.3   Mean   :2.982   Mean   :146.2  
 3rd Qu.: 333.2   3rd Qu.: 913.8   3rd Qu.:4.000   3rd Qu.:182.5  
 Max.   :1427.0   Max.   :4426.0   Max.   :5.000   Max.   :260.0

Korrelation zwischen den Prädiktoren

# Wir setzen die Schwelle bei |0.7|

cor <- cor(loyn[, 3:8]) # Korrelationen rechnen details siehe: "?cor"

# Korrelationen Visualisieren (google: "correlation plot r"...)
library("corrplot")

corrplot.mixed(cor, lower = "ellipse", upper = "number", order = "AOE")

-> Keine Korrelation ist >|0.7|, so können wir alle Prädiktoren “behalten”. Aber es gilt zu beachten , dass GRAZE ziemlich stark |>0.6| mit YR.ISOL korreliert ist

# Volles Modell definieren

names(loyn)

[1] "...1"    "ABUND"   "AREA"    "YR.ISOL" "DIST"    "LDIST"   "GRAZE"  
[8] "ALT"

lm.1 <- lm(ABUND ~ YR.ISOL + AREA + DIST + LDIST + GRAZE + ALT, data = loyn)

library("car")

par(mfrow = c(2, 2))
plot(lm.1)

-> Plot sieht zwar OK aus, aber mit 6 Prädiktoren für |<60| Beobachtungen ist das Modell wohl “overfitted”

# Andere Vatiante, um korrelierte Prädiktoren zu finden
vif(lm.1)

 YR.ISOL     AREA     DIST    LDIST    GRAZE      ALT 
1.841657 1.337627 1.227387 1.255028 2.307661 1.574537

Modellvereinfachung

Schrittweise die am wenigsten signifkanten Terme entfernen:

# Volles Modell anschauen
summary(lm.1)


Call:
lm(formula = ABUND ~ YR.ISOL + AREA + DIST + LDIST + GRAZE + 
    ALT, data = loyn)

Residuals:
     Min       1Q   Median       3Q      Max 
-17.6638  -4.6409  -0.0883   4.2858  20.1042 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept) -1.097e+02  1.133e+02  -0.968  0.33791   
YR.ISOL      6.693e-02  5.684e-02   1.177  0.24472   
AREA         8.866e-04  4.657e-03   0.190  0.84980   
DIST         3.811e-03  5.418e-03   0.703  0.48514   
LDIST        1.418e-03  1.310e-03   1.082  0.28451   
GRAZE       -3.447e+00  1.107e+00  -3.114  0.00308 **
ALT          4.772e-02  3.089e-02   1.545  0.12878   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.947 on 49 degrees of freedom
Multiple R-squared:  0.5118,    Adjusted R-squared:  0.452 
F-statistic: 8.561 on 6 and 49 DF,  p-value: 2.24e-06

lm.2 <- update(lm.1, ~ . - AREA) # Prädiktor mit grösstem p-Wert entfernen
anova(lm.1, lm.2) # Modelle vergleichen (falls signifikant, so müssten man den Prädiktor wieder ins Modell nehmen)

Analysis of Variance Table

Model 1: ABUND ~ YR.ISOL + AREA + DIST + LDIST + GRAZE + ALT
Model 2: ABUND ~ YR.ISOL + DIST + LDIST + GRAZE + ALT
  Res.Df    RSS Df Sum of Sq      F Pr(>F)
1     49 3094.2                           
2     50 3096.5 -1   -2.2886 0.0362 0.8498

summary(lm.2) # Neues einfacheres Modell anschauen und Prädiktor mit grösstem p-Wert ausfindig machen


Call:
lm(formula = ABUND ~ YR.ISOL + DIST + LDIST + GRAZE + ALT, data = loyn)

Residuals:
     Min       1Q   Median       3Q      Max 
-17.7240  -4.7245   0.0206   4.2698  20.0630 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept) -1.044e+02  1.089e+02  -0.959  0.34202   
YR.ISOL      6.418e-02  5.445e-02   1.179  0.24409   
DIST         3.884e-03  5.352e-03   0.726  0.47145   
LDIST        1.440e-03  1.292e-03   1.115  0.27036   
GRAZE       -3.500e+00  1.060e+00  -3.303  0.00177 **
ALT          4.964e-02  2.891e-02   1.717  0.09212 . 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.87 on 50 degrees of freedom
Multiple R-squared:  0.5114,    Adjusted R-squared:  0.4626 
F-statistic: 10.47 on 5 and 50 DF,  p-value: 6.532e-07

# Oben beschriebene Schritte wiederholen bis nur noch signifikante Prädiktoren im Modell
lm.3 <- update(lm.2, ~ . - DIST)
anova(lm.2, lm.3)

Analysis of Variance Table

Model 1: ABUND ~ YR.ISOL + DIST + LDIST + GRAZE + ALT
Model 2: ABUND ~ YR.ISOL + LDIST + GRAZE + ALT
  Res.Df    RSS Df Sum of Sq      F Pr(>F)
1     50 3096.5                           
2     51 3129.1 -1   -32.609 0.5265 0.4714

summary(lm.3)


Call:
lm(formula = ABUND ~ YR.ISOL + LDIST + GRAZE + ALT, data = loyn)

Residuals:
     Min       1Q   Median       3Q      Max 
-18.4659  -4.8236   0.1506   4.9245  19.8891 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -99.587487 108.158382  -0.921 0.361513    
YR.ISOL       0.062627   0.054157   1.156 0.252910    
LDIST         0.001677   0.001245   1.347 0.184026    
GRAZE        -3.699613   1.018706  -3.632 0.000653 ***
ALT           0.046485   0.028446   1.634 0.108386    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.833 on 51 degrees of freedom
Multiple R-squared:  0.5063,    Adjusted R-squared:  0.4676 
F-statistic: 13.07 on 4 and 51 DF,  p-value: 2.123e-07

lm.4 <- update(lm.3, ~ . - YR.ISOL)
anova(lm.3, lm.4)

Analysis of Variance Table

Model 1: ABUND ~ YR.ISOL + LDIST + GRAZE + ALT
Model 2: ABUND ~ LDIST + GRAZE + ALT
  Res.Df    RSS Df Sum of Sq      F Pr(>F)
1     51 3129.1                           
2     52 3211.2 -1   -82.047 1.3372 0.2529

summary(lm.4)


Call:
lm(formula = ABUND ~ LDIST + GRAZE + ALT, data = loyn)

Residuals:
    Min      1Q  Median      3Q     Max 
-18.155  -4.148  -0.503   4.649  18.588 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 25.289313   6.080034   4.159  0.00012 ***
LDIST        0.001455   0.001234   1.179  0.24362    
GRAZE       -4.430947   0.801206  -5.530 1.05e-06 ***
ALT          0.043565   0.028425   1.533  0.13144    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.858 on 52 degrees of freedom
Multiple R-squared:  0.4933,    Adjusted R-squared:  0.4641 
F-statistic: 16.88 on 3 and 52 DF,  p-value: 8.777e-08

lm.5 <- update(lm.4, ~ . - LDIST)
anova(lm.4, lm.5)

Analysis of Variance Table

Model 1: ABUND ~ LDIST + GRAZE + ALT
Model 2: ABUND ~ GRAZE + ALT
  Res.Df    RSS Df Sum of Sq      F Pr(>F)
1     52 3211.2                           
2     53 3297.1 -1   -85.892 1.3909 0.2436

summary(lm.5)


Call:
lm(formula = ABUND ~ GRAZE + ALT, data = loyn)

Residuals:
     Min       1Q   Median       3Q      Max 
-19.1677  -4.8261   0.0266   4.6944  19.1054 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 28.55582    5.43245   5.257 2.67e-06 ***
GRAZE       -4.59679    0.79167  -5.806 3.67e-07 ***
ALT          0.03191    0.02675   1.193    0.238    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.887 on 53 degrees of freedom
Multiple R-squared:  0.4798,    Adjusted R-squared:  0.4602 
F-statistic: 24.44 on 2 and 53 DF,  p-value: 3.011e-08

lm.6 <- update(lm.5, ~ . - ALT)
anova(lm.5, lm.6)

Analysis of Variance Table

Model 1: ABUND ~ GRAZE + ALT
Model 2: ABUND ~ GRAZE
  Res.Df    RSS Df Sum of Sq      F Pr(>F)
1     53 3297.1                           
2     54 3385.6 -1   -88.519 1.4229 0.2382

summary(lm.6)


Call:
lm(formula = ABUND ~ GRAZE, data = loyn)

Residuals:
     Min       1Q   Median       3Q      Max 
-19.1066  -5.4097   0.0934   4.4856  18.2747 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  34.3692     2.4095  14.264  < 2e-16 ***
GRAZE        -4.9813     0.7259  -6.862  6.9e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.918 on 54 degrees of freedom
Multiple R-squared:  0.4658,    Adjusted R-squared:  0.4559 
F-statistic: 47.09 on 1 and 54 DF,  p-value: 6.897e-09

par(mfrow = c(2, 2))
plot(lm.6)

-> das minimal adäquate Modell enthält nur noch einen Prädiktor (GRAZE) und dessen Residualplots sehen ok aus.

Hierarchical partitioning

Wir können auch schauen wie bedeutsam die einzelnen Variablen sind:

library("relaimpo")

# Lineares Modell definieren
lm_1 <- lm(ABUND~YR.ISOL+ AREA +  DIST + LDIST + GRAZE + ALT, data = loyn)

# Berechnen...
metrics <- calc.relimp(lm_1, type = c("lmg", "first", "last","betasq", "pratt"))
IJ <- cbind(I = metrics$lmg, J = metrics$first - metrics$lmg, Total = metrics$first)

IJ

                 I            J       Total
YR.ISOL 0.11827368  0.135095338 0.253369016
AREA    0.02359769  0.041923060 0.065520747
DIST    0.02566349  0.030085610 0.055749104
LDIST   0.01270789 -0.005112317 0.007595573
GRAZE   0.25879164  0.207030145 0.465821782
ALT     0.07275743  0.076112117 0.148869552

-> auch hier sticht GRAZE heraus. (und an zweiter Stelle YR.ISOL, der mit GRAZE am stärksten korreliert ist)

Partial regressions

avPlots(lm.1, ask = F)

Multimodel inference

library("MuMIn")

global.model <- lm(ABUND ~ YR.ISOL + AREA +  DIST + LDIST + GRAZE + ALT, data = loyn)

options(na.action = "na.fail")

allmodels <- dredge(global.model)
allmodels

Global model call: lm(formula = ABUND ~ YR.ISOL + AREA + DIST + LDIST + GRAZE + 
    ALT, data = loyn)
---
Model selection table 
       (Int)     ALT        ARE      DIS    GRA       LDI  YR.ISO df   logLik
9    34.3700                             -4.981                    3 -194.315
10   28.5600 0.03191                     -4.597                    4 -193.573
41  -62.7500                             -4.440           0.04898  4 -193.886
26   25.2900 0.04356                     -4.431 0.0014550          5 -192.834
25   33.7400                             -4.967 0.0007979          4 -194.071
13   33.2300                    0.003224 -4.858                    4 -194.102
11   33.9000          1.969e-03          -4.871                    4 -194.201
14   25.6300 0.03834            0.004970 -4.330                    5 -193.081
42  -73.5800 0.03285                     -4.017           0.05143  5 -193.087
58  -99.5900 0.04649                     -3.700 0.0016770 0.06263  6 -192.109
57  -74.9700                             -4.359 0.0009527 0.05477  5 -193.538
12   28.6100 0.03094  5.055e-04          -4.580                    5 -193.566
43  -85.8200          3.243e-03          -4.133           0.06023  5 -193.596
45  -68.9900                    0.003542 -4.277           0.05150  5 -193.626
46  -85.3900 0.03990            0.005386 -3.679           0.05577  6 -192.503
30   23.6500 0.04645            0.003635 -4.262 0.0012290          6 -192.583
27   33.3100          1.892e-03          -4.861 0.0007836          5 -193.965
29   33.0400                    0.002360 -4.880 0.0006227          5 -193.968
15   32.8100          1.886e-03 0.003153 -4.755                    5 -193.997
28   25.2400 0.04416 -2.645e-04          -4.438 0.0014660          6 -192.832
44  -85.1900 0.02954  1.785e-03          -3.891           0.05737  6 -193.006
16   25.6300 0.03828  2.748e-05 0.004967 -4.329                    6 -193.081
62 -104.4000 0.04964            0.003884 -3.500 0.0014400 0.06418  7 -191.816
59  -98.3600          3.274e-03          -4.049 0.0009602 0.06617  6 -193.239
47  -91.6700          3.200e-03 0.003490 -3.977           0.06256  6 -193.341
60 -106.3000 0.04414  1.117e-03          -3.627 0.0016430 0.06612  7 -192.076
61  -77.0200                    0.002499 -4.260 0.0007691 0.05543  6 -193.420
48  -93.8000 0.03722  1.342e-03 0.005240 -3.593           0.06013  7 -192.456
31   32.6300          1.848e-03 0.002304 -4.779 0.0006130          6 -193.867
32   23.5300 0.04762 -5.038e-04 0.003682 -4.274 0.0012470          7 -192.576
63 -100.1000          3.239e-03 0.002430 -3.956 0.0007817 0.06669  7 -193.127
64 -109.7000 0.04772  8.866e-04 0.003811 -3.447 0.0014180 0.06693  8 -191.795
38 -325.3000 0.07807            0.011030                  0.16960  5 -198.542
50 -355.4000 0.08742                            0.0027220 0.18470  5 -198.549
54 -336.0000 0.08950            0.008464        0.0020850 0.17380  6 -197.343
52 -363.9000 0.07191  5.500e-03                 0.0024540 0.19020  6 -197.858
40 -336.7000 0.06363  5.431e-03 0.009912                  0.17650  6 -197.870
56 -344.8000 0.07638  4.590e-03 0.007723        0.0019170 0.17930  7 -196.852
34 -348.5000 0.07006                                      0.18350  4 -200.670
36 -360.2000 0.05243  7.028e-03                           0.19060  5 -199.584
35 -393.4000          1.036e-02                           0.21140  4 -201.103
39 -380.7000          9.678e-03 0.007598                  0.20400  5 -200.121
51 -402.6000          1.019e-02                 0.0014210 0.21560  5 -200.496
55 -389.0000          9.682e-03 0.006312        0.0009292 0.20800  6 -199.883
37 -377.6000                    0.008890                  0.20260  4 -202.444
33 -392.3000                                              0.21120  3 -203.690
49 -402.3000                                    0.0015230 0.21580  4 -203.054
53 -385.8000                    0.007611        0.0009246 0.20660  5 -202.227
6     1.1570 0.10280            0.013820                           4 -204.646
22   -1.1380 0.11310            0.011680        0.0017790          5 -203.948
8     2.2060 0.09511  3.112e-03 0.013240                           5 -204.467
18    1.1530 0.11220                            0.0026530          4 -205.786
2     5.5980 0.09515                                               3 -207.358
24   -0.2323 0.10680  2.339e-03 0.011360        0.0016890          6 -203.846
20    2.4160 0.10280  3.509e-03                 0.0024810          5 -205.568
4     6.9990 0.08318  5.050e-03                                    4 -206.917
7    16.3800          9.404e-03 0.010330                           4 -208.625
3    18.8000          1.033e-02                                    3 -209.974
5    16.7300                    0.011570                           3 -210.265
1    19.5100                                                       2 -211.871
19   18.1300          1.022e-02                 0.0009186          4 -209.789
23   16.3100          9.404e-03 0.010120        0.0001591          5 -208.620
21   16.6700                    0.011360        0.0001598          4 -210.260
17   18.7700                                    0.0010210          3 -211.658
    AICc delta weight
9  395.1  0.00  0.145
10 395.9  0.84  0.095
41 396.6  1.46  0.070
26 396.9  1.78  0.059
25 396.9  1.84  0.058
13 397.0  1.90  0.056
11 397.2  2.10  0.051
14 397.4  2.27  0.046
42 397.4  2.28  0.046
58 397.9  2.84  0.035
57 398.3  3.19  0.029
12 398.3  3.24  0.029
43 398.4  3.30  0.028
45 398.5  3.36  0.027
46 398.7  3.63  0.024
30 398.9  3.79  0.022
27 399.1  4.04  0.019
29 399.1  4.04  0.019
15 399.2  4.10  0.019
28 399.4  4.29  0.017
44 399.7  4.63  0.014
16 399.9  4.79  0.013
62 400.0  4.87  0.013
59 400.2  5.10  0.011
47 400.4  5.31  0.010
60 400.5  5.40  0.010
61 400.6  5.46  0.009
48 401.2  6.15  0.007
31 401.4  6.36  0.006
32 401.5  6.39  0.006
63 402.6  7.50  0.003
64 402.7  7.56  0.003
38 408.3 13.19  0.000
50 408.3 13.21  0.000
54 408.4 13.31  0.000
52 409.4 14.34  0.000
40 409.5 14.36  0.000
56 410.0 14.95  0.000
34 410.1 15.03  0.000
36 410.4 15.28  0.000
35 411.0 15.90  0.000
39 411.4 16.35  0.000
51 412.2 17.10  0.000
55 413.5 18.39  0.000
37 413.7 18.58  0.000
33 413.8 18.75  0.000
49 414.9 19.80  0.000
53 415.7 20.56  0.000
6  418.1 22.99  0.000
22 419.1 24.01  0.000
8  420.1 25.04  0.000
18 420.4 25.27  0.000
2  421.2 26.09  0.000
24 421.4 26.32  0.000
20 422.3 27.25  0.000
4  422.6 27.53  0.000
7  426.0 30.94  0.000
3  426.4 31.32  0.000
5  427.0 31.90  0.000
1  428.0 32.88  0.000
19 428.4 33.27  0.000
23 428.4 33.35  0.000
21 429.3 34.21  0.000
17 429.8 34.69  0.000
Models ranked by AICc(x)

# Variable importance
sw(allmodels)

                     GRAZE ALT  YR.ISOL LDIST DIST AREA
Sum of weights:      1.00  0.44 0.34    0.32  0.28 0.25
N containing models:   32    32   32      32    32   32

-> Auch mit dieser Sichtweise ist GRAZE der wichtigste Prädiktor

# Model averaging
avgmodel <- model.avg(allmodels, subset = TRUE)
summary(avgmodel)


Call:
model.avg(object = allmodels, subset = TRUE)

Component model call: 
lm(formula = ABUND ~ <64 unique rhs>, data = loyn)

Component models: 
       df  logLik   AICc delta weight
4       3 -194.31 395.09  0.00   0.14
14      4 -193.57 395.93  0.84   0.10
46      4 -193.89 396.56  1.46   0.07
145     5 -192.83 396.87  1.78   0.06
45      4 -194.07 396.93  1.84   0.06
34      4 -194.10 396.99  1.90   0.06
24      4 -194.20 397.19  2.10   0.05
134     5 -193.08 397.36  2.27   0.05
146     5 -193.09 397.37  2.28   0.05
1456    6 -192.11 397.93  2.84   0.03
456     5 -193.54 398.28  3.19   0.03
124     5 -193.57 398.33  3.24   0.03
246     5 -193.60 398.39  3.30   0.03
346     5 -193.63 398.45  3.36   0.03
1346    6 -192.50 398.72  3.63   0.02
1345    6 -192.58 398.88  3.79   0.02
245     5 -193.97 399.13  4.04   0.02
345     5 -193.97 399.14  4.04   0.02
234     5 -194.00 399.19  4.10   0.02
1245    6 -192.83 399.38  4.29   0.02
1246    6 -193.01 399.73  4.63   0.01
1234    6 -193.08 399.88  4.79   0.01
13456   7 -191.82 399.96  4.87   0.01
2456    6 -193.24 400.19  5.10   0.01
2346    6 -193.34 400.40  5.31   0.01
12456   7 -192.08 400.49  5.40   0.01
3456    6 -193.42 400.55  5.46   0.01
12346   7 -192.46 401.25  6.15   0.01
2345    6 -193.87 401.45  6.36   0.01
12345   7 -192.58 401.49  6.39   0.01
23456   7 -193.13 402.59  7.50   0.00
123456  8 -191.79 402.65  7.56   0.00
136     5 -198.54 408.28 13.19   0.00
156     5 -198.55 408.30 13.21   0.00
1356    6 -197.34 408.40 13.31   0.00
1256    6 -197.86 409.43 14.34   0.00
1236    6 -197.87 409.45 14.36   0.00
12356   7 -196.85 410.04 14.95   0.00
16      4 -200.67 410.13 15.03   0.00
126     5 -199.58 410.37 15.28   0.00
26      4 -201.10 410.99 15.90   0.00
236     5 -200.12 411.44 16.35   0.00
256     5 -200.50 412.19 17.10   0.00
2356    6 -199.88 413.48 18.39   0.00
36      4 -202.44 413.67 18.58   0.00
6       3 -203.69 413.84 18.75   0.00
56      4 -203.05 414.89 19.80   0.00
356     5 -202.23 415.65 20.56   0.00
13      4 -204.65 418.08 22.99   0.00
135     5 -203.95 419.10 24.01   0.00
123     5 -204.47 420.13 25.04   0.00
15      4 -205.79 420.36 25.27   0.00
1       3 -207.36 421.18 26.09   0.00
1235    6 -203.85 421.41 26.32   0.00
125     5 -205.57 422.34 27.25   0.00
12      4 -206.92 422.62 27.53   0.00
23      4 -208.63 426.04 30.94   0.00
2       3 -209.97 426.41 31.32   0.00
3       3 -210.27 426.99 31.90   0.00
(Null)  2 -211.87 427.97 32.88   0.00
25      4 -209.79 428.36 33.27   0.00
235     5 -208.62 428.44 33.35   0.00
35      4 -210.26 429.30 34.21   0.00
5       3 -211.66 429.78 34.69   0.00

Term codes: 
    ALT    AREA    DIST   GRAZE   LDIST YR.ISOL 
      1       2       3       4       5       6 

Model-averaged coefficients:  
(full average) 
              Estimate Std. Error Adjusted SE z value Pr(>|z|)    
(Intercept) -7.5601112 84.0460963  85.2129010   0.089    0.929    
GRAZE       -4.4923671  0.9653718   0.9836120   4.567  4.9e-06 ***
ALT          0.0168833  0.0269646   0.0272737   0.619    0.536    
YR.ISOL      0.0192028  0.0420833   0.0426707   0.450    0.653    
LDIST        0.0003700  0.0009029   0.0009158   0.404    0.686    
DIST         0.0010880  0.0033129   0.0033689   0.323    0.747    
AREA         0.0004120  0.0023720   0.0024205   0.170    0.865    
 
(conditional average) 
             Estimate Std. Error Adjusted SE z value Pr(>|z|)    
(Intercept) -7.560111  84.046096   85.212901   0.089    0.929    
GRAZE       -4.497801   0.953221    0.971711   4.629  3.7e-06 ***
ALT          0.038392   0.028768    0.029424   1.305    0.192    
YR.ISOL      0.056407   0.055710    0.057008   0.989    0.322    
LDIST        0.001152   0.001280    0.001308   0.881    0.378    
DIST         0.003833   0.005305    0.005428   0.706    0.480    
AREA         0.001673   0.004554    0.004656   0.359    0.719    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1