Generalised linear models

• Overview
• Types
• Assumptions
• Fitting
• Examples

Overview

• GLMs are a broad class of models for fitting different types of response variables distributions.
• The multiple linear regression model is a special case.

Three components

• Random Component: probability distribution of the response variable
• Systematic Component: explanatory variables
• Link function: describes the relaionship between the random and systematic components

Multiple linear regression

$$y_i={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\epsilon orE\left(Y_i\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2$$

• Random component: $y_i$ has a normal distribution, and so $e_i\sim N(0,{\sigma }^2)$
• Systematic component: ${\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2$
• Link function: identity, just the systematic component

Poisson regression

$$y_i=exp({\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2)+\epsilon$$

• $y_i$ takes integer values, 0, 1, 2, ...
• Link function: $ln\left(\mu \right)$, name=log. (Think of $\mu$ as $\hat{y}$.)

Bernouilli, binomial regression

$$y_i=\frac{exp({\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2)}{1+exp({\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2)}+\epsilon$$

• $y_i$ takes integer values, $\{0,1\}$ (bernouilli), $\{0,1,...,n\}$ (binomial)
• Let $\mu =\frac{exp({\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2)}{1+exp({\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2)}$, link function is $ln\frac{\mu }{1-\mu }$, name=logit

Assumptions

• The data $y_1,y_2,...,y_n$ are independently distributed, i.e., cases are independent.
• The dependent variable $y_i$ does NOT need to be normally distributed, but it typically assumes a distribution from an exponential family (e.g. binomial, Poisson, multinomial, normal,...)
• Linear relationship between the transformed response (see examples below)
• Explanatory variables can be transformations of original variables
• Homogeneity of variance does NOT need to be satisfied for original units, but it should be still true on the transformed response scale
• Uses maximum likelihood estimation (MLE) to estimate the parameters
• Goodness-of-fit measures rely on sufficiently large samples

Example: Olympics medal tally

• Model medal counts on log_GDP
• Medal counts = integer, which suggests using a Poisson model.

Model fit and what it looks like

oly_glm <- glm(M2012~GDP_log, data=oly_gdp2012,
               family=poisson(link=log))
summary(oly_glm)$coefficients
#>             Estimate Std. Error z value Pr(>|z|)
#> (Intercept)    -13.2      0.538     -24 3.6e-132
#> GDP_log          1.3      0.045      30 6.8e-198

Model fit

#> 
#> Call:
#> glm(formula = M2012 ~ GDP_log, family = poisson(link = log), 
#>     data = oly_gdp2012)
#> 
#> Deviance Residuals: 
#>    Min      1Q  Median      3Q     Max  
#>  -4.80   -2.22   -0.36    1.07    8.55  
#> 
#> Coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept) -13.1691     0.5383   -24.5   <2e-16 ***
#> GDP_log       1.3406     0.0447    30.0   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for poisson family taken to be 1)
#> 
#>     Null deviance: 1567.70  on 84  degrees of freedom
#> Residual deviance:  545.92  on 83  degrees of freedom
#> AIC: 845.7
#> 
#> Number of Fisher Scoring iterations: 5

The difference between the null and residual deviance is substantial, suggesting a good fit.

Residual plots

Heteroskedasticity in residuals. One fairly large residual.

Influence statistics

#>         .rownames .cooksd .resid
#> 1      RussianFed 1.9e+00  8.553
#> 2           China 1.5e+00  3.743
#> 3    UnitedStates 8.3e-01  1.468
#> 4    GreatBritain 8.0e-01  5.232
#> 5         Jamaica 4.4e-01  5.267
#> 6           India 2.6e-01 -4.800
#> 7           Japan 2.5e-01 -2.010
#> 8            Cuba 2.4e-01  4.215
#> 9         Ukraine 2.3e-01  4.270
#> 10          Kenya 1.9e-01  3.802
#> 11        Belarus 1.6e-01  3.535
#> 12        Hungary 1.5e-01  3.621
#> 13         Brazil 1.5e-01 -2.862
#> 14        Georgia 1.3e-01  3.219
#> 15      Indonesia 1.2e-01 -4.563
#> 16         Mexico 9.8e-02 -3.444
#> 17    SaudiArabia 9.2e-02 -4.388
#> 18      Australia 7.6e-02  2.211
#> 19     Azerbaijan 7.5e-02  2.584
#> 20       Mongolia 7.3e-02  2.612
#> 21  ChineseTaipei 7.0e-02 -3.680
#> 22         Turkey 6.5e-02 -3.179
#> 23    Switzerland 6.5e-02 -3.293
#> 24       Ethiopia 6.2e-02  2.385
#> 25        Belgium 6.0e-02 -3.294
#> 26      Venezuela 5.8e-02 -3.498
#> 27     NewZealand 5.0e-02  2.211
#> 28  HongKongChina 4.9e-02 -3.191
#> 29       Portugal 4.9e-02 -3.164
#> 30         Greece 4.5e-02 -2.932
#> 31     Kazakhstan 4.4e-02  2.100
#> 32         Norway 4.3e-02 -2.700
#> 33       DPRKorea 4.2e-02  2.020
#> 34        Algeria 4.0e-02 -2.815
#> 35      Singapore 3.9e-02 -2.705
#> 36      Argentina 3.8e-02 -2.534
#> 37         Kuwait 3.8e-02 -2.731
#> 38       Thailand 3.7e-02 -2.566
#> 39       Malaysia 3.7e-02 -2.602
#> 40         Canada 3.6e-02 -1.607
#> 41          Egypt 3.4e-02 -2.512
#> 42          Korea 3.3e-02  1.635
#> 43        Finland 2.9e-02 -2.222
#> 44          Spain 2.6e-02 -1.463
#> 45          Qatar 2.6e-02 -2.126
#> 46        Morocco 2.4e-02 -2.147
#> 47        Germany 2.1e-02  0.754
#> 48    SouthAfrica 1.9e-02 -1.705
#> 49         Sweden 1.8e-02 -1.586
#> 50        Armenia 1.4e-02  1.291
#> 51 TrinidadTobago 1.4e-02  1.234
#> 52     PuertoRico 1.2e-02 -1.390
#> 53      Guatemala 1.1e-02 -1.396
#> 54        Croatia 1.1e-02  1.073
#> 55      Lithuania 1.0e-02  1.072
#> 56        Ireland 7.5e-03 -1.044
#> 57  CzechRepublic 5.5e-03  0.804
#> 58        Grenada 5.2e-03  1.025
#> 59    Netherlands 5.2e-03  0.726
#> 60         Poland 5.0e-03 -0.817
#> 61  Rep.ofMoldova 4.9e-03  0.827
#> 62        Romania 4.8e-03  0.750
#> 63        Bahrain 4.8e-03 -0.925
#> 64         Cyprus 4.6e-03 -0.904
#> 65  DominicanRep. 4.4e-03 -0.822
#> 66       Bulgaria 4.4e-03 -0.820
#> 67     Uzbekistan 2.6e-03  0.555
#> 68         Serbia 2.5e-03  0.550
#> 69    Afghanistan 2.3e-03 -0.637
#> 70       Colombia 2.0e-03 -0.518
#> 71          Gabon 1.9e-03 -0.588
#> 72       Botswana 1.9e-03 -0.575
#> 73          Italy 1.8e-03 -0.304
#> 74         Uganda 1.7e-03 -0.558
#> 75       Slovenia 1.1e-03  0.359
#> 76       Slovakia 9.6e-04 -0.360
#> 77        Denmark 8.2e-04 -0.330
#> 78     Montenegro 3.6e-04  0.257
#> 79         Latvia 2.4e-04 -0.189
#> 80        Tunisia 8.8e-05 -0.109
#> 81        Bahamas 7.5e-05 -0.116
#> 82         France 4.7e-05  0.042
#> 83           Iran 3.5e-06 -0.021
#> 84        Estonia 3.5e-06 -0.022
#> 85     Tajikistan 8.3e-07 -0.012

Largest Cooks D values enough to have some concerns about the influence that Russian Federation and China have on the model fit. Should re-fit without these two cases.

Prediction from the model

aus <- oly_gdp2012 %>% filter(Code == "AUS")
predict(oly_glm, aus)
#>   1 
#> 3.2

WAIT! What??? Australia earned more than 3 medals in 2012. Either the model is terrible, or we've made a mistake!

Prediction from the model

aus <- oly_gdp2012 %>% filter(Code == "AUS")
predict(oly_glm, aus)
#>   1 
#> 3.2

WAIT! What??? Australia earned more than 3 medals in 2012. Either the model is terrible, or we've made a mistake!

aus <- oly_gdp2012 %>% filter(Code == "AUS")
predict(oly_glm, aus, type="response")
#>  1 
#> 23

Prediction from the model

aus <- oly_gdp2012 %>% filter(Code == "AUS")
predict(oly_glm, aus)
#>   1 
#> 3.2

WAIT! What??? Australia earned more than 3 medals in 2012. Either the model is terrible, or we've made a mistake!

aus <- oly_gdp2012 %>% filter(Code == "AUS")
predict(oly_glm, aus, type="response")
#>  1 
#> 23

Need to transform predictions into original units.

Example: winning tennis matches

We have data scraped from the web sites of the 2012 Grand Slam tennis tournaments. There are a lot of statistics on matches. Below we have the number of receiving points won, and whether the match was won or not.

Model

tennis_glm <- glm(won~Receiving.Points.Won, data=tennis,
                  family=binomial(link='logit'))

#>                      Estimate Std. Error z value Pr(>|z|)
#> (Intercept)             -2.91      0.586    -5.0  7.1e-07
#> Receiving.Points.Won     0.11      0.015     7.3  3.0e-13

Model fit

#> 
#> Call:
#> glm(formula = won ~ Receiving.Points.Won, family = binomial(link = "logit"), 
#>     data = tennis)
#> 
#> Deviance Residuals: 
#>    Min      1Q  Median      3Q     Max  
#> -2.506   0.227   0.411   0.624   1.877  
#> 
#> Coefficients:
#>                      Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)           -2.9053     0.5860   -4.96  7.1e-07 ***
#> Receiving.Points.Won   0.1111     0.0152    7.29  3.0e-13 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 472.99  on 511  degrees of freedom
#> Residual deviance: 402.16  on 510  degrees of freedom
#> AIC: 406.2
#> 
#> Number of Fisher Scoring iterations: 5

Not much difference between null and residual deviance, suggests return points won does not explain much of the match result.

Residuals

Model is just not capturing the data very well. There are two groups of residuals, its overfitting a chunk and underfitting chunks of data.

Influence statistics

#>     .cooksd .resid
#> 1   6.0e-02  1.877
#> 2   3.6e-02 -2.505
#> 3   2.9e-02 -2.420
#> 4   2.4e-02  1.528
#> 5   2.0e-02 -2.287
#> 6   1.7e-02 -2.242
#> 7   1.7e-02 -2.242
#> 8   1.5e-02 -2.196
#> 9   1.3e-02 -2.149
#> 10  1.2e-02  1.329
#> 11  1.2e-02  1.329
#> 12  1.1e-02 -2.103
#> 13  1.1e-02 -2.103
#> 14  1.1e-02 -2.103
#> 15  9.9e-03 -2.055
#> 16  9.9e-03 -2.055
#> 17  9.9e-03 -2.055
#> 18  9.9e-03 -2.055
#> 19  9.4e-03  1.280
#> 20  9.4e-03  1.280
#> 21  9.4e-03  1.280
#> 22  9.4e-03  1.280
#> 23  8.6e-03 -2.008
#> 24  7.6e-03  1.232
#> 25  7.6e-03  1.232
#> 26  7.5e-03 -1.959
#> 27  7.5e-03 -1.959
#> 28  7.5e-03 -1.959
#> 29  7.5e-03 -1.959
#> 30  7.5e-03 -1.959
#> 31  7.5e-03 -1.959
#> 32  6.6e-03 -1.911
#> 33  6.6e-03 -1.911
#> 34  5.9e-03 -1.124
#> 35  5.9e-03 -1.170
#> 36  5.9e-03 -1.862
#> 37  5.9e-03 -1.862
#> 38  5.9e-03 -1.078
#> 39  5.9e-03 -1.078
#> 40  5.7e-03 -1.266
#> 41  5.7e-03 -1.266
#> 42  5.6e-03 -1.315
#> 43  5.6e-03 -1.315
#> 44  5.6e-03 -1.315
#> 45  5.6e-03 -1.315
#> 46  5.6e-03 -1.315
#> 47  5.6e-03 -1.315
#> 48  5.6e-03 -0.989
#> 49  5.4e-03 -1.364
#> 50  5.4e-03 -1.364
#> 51  5.4e-03 -1.364
#> 52  5.4e-03 -1.364
#> 53  5.4e-03 -0.946
#> 54  5.3e-03 -1.813
#> 55  5.3e-03 -1.813
#> 56  5.3e-03 -1.813
#> 57  5.2e-03 -1.413
#> 58  5.2e-03 -1.413
#> 59  5.2e-03 -1.413
#> 60  5.2e-03 -1.413
#> 61  5.2e-03 -1.413
#> 62  5.2e-03 -1.413
#> 63  5.2e-03 -1.413
#> 64  5.0e-03 -1.463
#> 65  5.0e-03 -1.463
#> 66  5.0e-03 -1.463
#> 67  5.0e-03 -1.463
#> 68  5.0e-03 -1.463
#> 69  5.0e-03 -1.463
#> 70  5.0e-03 -1.763
#> 71  5.0e-03 -1.763
#> 72  5.0e-03 -1.763
#> 73  5.0e-03 -1.763
#> 74  5.0e-03 -1.763
#> 75  4.9e-03 -1.513
#> 76  4.9e-03 -1.513
#> 77  4.9e-03 -1.513
#> 78  4.9e-03 -1.513
#> 79  4.9e-03 -1.513
#> 80  4.9e-03 -1.513
#> 81  4.9e-03 -1.513
#> 82  4.8e-03  1.138
#> 83  4.8e-03  1.138
#> 84  4.8e-03  1.138
#> 85  4.8e-03  1.138
#> 86  4.8e-03  1.138
#> 87  4.8e-03 -1.713
#> 88  4.8e-03 -1.713
#> 89  4.8e-03 -1.713
#> 90  4.8e-03 -1.713
#> 91  4.7e-03 -1.563
#> 92  4.7e-03 -1.563
#> 93  4.7e-03 -1.563
#> 94  4.6e-03 -1.663
#> 95  4.6e-03 -1.663
#> 96  4.6e-03 -1.663
#> 97  4.6e-03 -1.663
#> 98  4.6e-03 -1.613
#> 99  4.6e-03 -1.613
#> 100 4.6e-03 -1.613
#> 101 4.6e-03 -1.613
#> 102 4.6e-03 -1.613
#> 103 4.6e-03 -1.613
#> 104 3.8e-03  1.091
#> 105 3.8e-03  1.091
#> 106 3.8e-03  1.091
#> 107 3.0e-03  1.046
#> 108 3.0e-03  1.046
#> 109 3.0e-03  1.046
#> 110 2.6e-03 -0.614
#> 111 2.3e-03  1.002
#> 112 2.3e-03  1.002
#> 113 2.3e-03  1.002
#> 114 2.3e-03  1.002
#> 115 2.3e-03  1.002
#> 116 2.3e-03  1.002
#> 117 2.3e-03  1.002
#> 118 1.8e-03  0.959
#> 119 1.8e-03  0.959
#> 120 1.8e-03  0.959
#> 121 1.8e-03  0.959
#> 122 1.8e-03  0.959
#> 123 1.8e-03  0.959
#> 124 1.8e-03  0.959
#> 125 1.8e-03  0.959
#> 126 1.8e-03  0.959
#> 127 1.8e-03  0.959
#> 128 1.8e-03  0.959
#> 129 1.4e-03  0.917
#> 130 1.4e-03  0.917
#> 131 1.4e-03  0.917
#> 132 1.4e-03  0.917
#> 133 1.4e-03  0.917
#> 134 1.4e-03  0.917
#> 135 1.4e-03  0.917
#> 136 1.4e-03  0.917
#> 137 1.4e-03  0.917
#> 138 1.4e-03  0.917
#> 139 1.4e-03  0.917
#> 140 1.1e-03  0.876
#> 141 1.1e-03  0.876
#> 142 1.1e-03  0.876
#> 143 1.1e-03  0.876
#> 144 1.1e-03  0.876
#> 145 1.1e-03  0.876
#> 146 1.1e-03  0.876
#> 147 8.3e-04  0.836
#> 148 8.3e-04  0.836
#> 149 8.3e-04  0.836
#> 150 8.3e-04  0.836
#> 151 8.3e-04  0.836
#> 152 8.3e-04  0.836
#> 153 8.3e-04  0.836
#> 154 6.5e-04  0.797
#> 155 6.5e-04  0.797
#> 156 6.5e-04  0.797
#> 157 6.5e-04  0.797
#> 158 6.5e-04  0.797
#> 159 6.5e-04  0.797
#> 160 6.5e-04  0.797
#> 161 6.5e-04  0.797
#> 162 6.5e-04  0.797
#> 163 6.5e-04  0.797
#> 164 6.5e-04  0.797
#> 165 6.5e-04  0.797
#> 166 6.5e-04  0.797
#> 167 6.5e-04  0.797
#> 168 6.5e-04  0.797
#> 169 5.2e-04  0.760
#> 170 5.2e-04  0.760
#> 171 5.2e-04  0.760
#> 172 5.2e-04  0.760
#> 173 5.2e-04  0.760
#> 174 5.2e-04  0.760
#> 175 5.2e-04  0.760
#> 176 5.2e-04  0.760
#> 177 5.2e-04  0.760
#> 178 5.2e-04  0.760
#> 179 5.2e-04  0.760
#> 180 5.2e-04  0.760
#> 181 4.3e-04  0.724
#> 182 4.3e-04  0.724
#> 183 4.3e-04  0.724
#> 184 4.3e-04  0.724
#> 185 4.3e-04  0.724
#> 186 4.3e-04  0.724
#> 187 4.3e-04  0.724
#> 188 4.3e-04  0.724
#> 189 4.3e-04  0.724
#> 190 4.3e-04  0.724
#> 191 4.3e-04  0.724
#> 192 4.3e-04  0.724
#> 193 4.3e-04  0.724
#> 194 4.3e-04  0.724
#> 195 3.6e-04  0.689
#> 196 3.6e-04  0.689
#> 197 3.6e-04  0.689
#> 198 3.6e-04  0.689
#> 199 3.6e-04  0.689
#> 200 3.6e-04  0.689
#> 201 3.6e-04  0.689
#> 202 3.6e-04  0.689
#> 203 3.6e-04  0.689
#> 204 3.6e-04  0.689
#> 205 3.6e-04  0.689
#> 206 3.1e-04  0.656
#> 207 3.1e-04  0.656
#> 208 3.1e-04  0.656
#> 209 3.1e-04  0.656
#> 210 3.1e-04  0.656
#> 211 3.1e-04  0.656
#> 212 3.1e-04  0.656
#> 213 3.1e-04  0.656
#> 214 2.7e-04  0.624
#> 215 2.7e-04  0.624
#> 216 2.7e-04  0.624
#> 217 2.7e-04  0.624
#> 218 2.7e-04  0.624
#> 219 2.7e-04  0.624
#> 220 2.7e-04  0.624
#> 221 2.4e-04  0.593
#> 222 2.4e-04  0.593
#> 223 2.4e-04  0.593
#> 224 2.4e-04  0.593
#> 225 2.4e-04  0.593
#> 226 2.4e-04  0.593
#> 227 2.4e-04  0.593
#> 228 2.4e-04  0.593
#> 229 2.4e-04  0.593
#> 230 2.4e-04  0.593
#> 231 2.4e-04  0.593
#> 232 2.4e-04  0.593
#> 233 2.4e-04  0.593
#> 234 2.4e-04  0.593
#> 235 2.4e-04  0.593
#> 236 2.2e-04  0.563
#> 237 2.2e-04  0.563
#> 238 2.2e-04  0.563
#> 239 2.2e-04  0.563
#> 240 2.2e-04  0.563
#> 241 2.2e-04  0.563
#> 242 2.2e-04  0.563
#> 243 2.2e-04  0.563
#> 244 2.2e-04  0.563
#> 245 2.2e-04  0.563
#> 246 2.2e-04  0.563
#> 247 2.2e-04  0.563
#> 248 2.2e-04  0.563
#> 249 2.2e-04  0.563
#> 250 2.2e-04  0.563
#> 251 2.2e-04  0.563
#> 252 2.2e-04  0.563
#> 253 2.2e-04  0.563
#> 254 2.2e-04  0.563
#> 255 2.0e-04  0.535
#> 256 2.0e-04  0.535
#> 257 2.0e-04  0.535
#> 258 2.0e-04  0.535
#> 259 2.0e-04  0.535
#> 260 2.0e-04  0.535
#> 261 2.0e-04  0.535
#> 262 2.0e-04  0.535
#> 263 2.0e-04  0.535
#> 264 2.0e-04  0.535
#> 265 2.0e-04  0.535
#> 266 2.0e-04  0.535
#> 267 2.0e-04  0.535
#> 268 2.0e-04  0.535
#> 269 2.0e-04  0.535
#> 270 2.0e-04  0.535
#> 271 2.0e-04  0.535
#> 272 1.9e-04  0.508
#> 273 1.9e-04  0.508
#> 274 1.9e-04  0.508
#> 275 1.9e-04  0.508
#> 276 1.9e-04  0.508
#> 277 1.9e-04  0.508
#> 278 1.9e-04  0.508
#> 279 1.9e-04  0.508
#> 280 1.9e-04  0.508
#> 281 1.9e-04  0.508
#> 282 1.9e-04  0.508
#> 283 1.9e-04  0.508
#> 284 1.9e-04  0.508
#> 285 1.9e-04  0.508
#> 286 1.9e-04  0.508
#> 287 1.9e-04  0.508
#> 288 1.9e-04  0.508
#> 289 1.9e-04  0.508
#> 290 1.7e-04  0.482
#> 291 1.7e-04  0.482
#> 292 1.7e-04  0.482
#> 293 1.7e-04  0.482
#> 294 1.7e-04  0.482
#> 295 1.7e-04  0.482
#> 296 1.7e-04  0.482
#> 297 1.7e-04  0.482
#> 298 1.7e-04  0.482
#> 299 1.7e-04  0.482
#> 300 1.7e-04  0.482
#> 301 1.7e-04  0.482
#> 302 1.7e-04  0.482
#> 303 1.7e-04  0.482
#> 304 1.7e-04  0.482
#> 305 1.7e-04  0.482
#> 306 1.7e-04  0.482
#> 307 1.7e-04  0.482
#> 308 1.7e-04  0.482
#> 309 1.7e-04  0.482
#> 310 1.7e-04  0.482
#> 311 1.7e-04  0.482
#> 312 1.7e-04  0.482
#> 313 1.6e-04  0.457
#> 314 1.6e-04  0.457
#> 315 1.6e-04  0.457
#> 316 1.6e-04  0.457
#> 317 1.6e-04  0.457
#> 318 1.6e-04  0.457
#> 319 1.6e-04  0.457
#> 320 1.6e-04  0.457
#> 321 1.6e-04  0.457
#> 322 1.6e-04  0.457
#> 323 1.6e-04  0.457
#> 324 1.6e-04  0.457
#> 325 1.6e-04  0.457
#> 326 1.5e-04  0.434
#> 327 1.5e-04  0.434
#> 328 1.5e-04  0.434
#> 329 1.5e-04  0.434
#> 330 1.5e-04  0.434
#> 331 1.5e-04  0.434
#> 332 1.5e-04  0.434
#> 333 1.5e-04  0.434
#> 334 1.5e-04  0.434
#> 335 1.5e-04  0.434
#> 336 1.5e-04  0.434
#> 337 1.5e-04  0.434
#> 338 1.5e-04  0.434
#> 339 1.5e-04  0.434
#> 340 1.5e-04  0.434
#> 341 1.5e-04  0.434
#> 342 1.5e-04  0.434
#> 343 1.4e-04  0.411
#> 344 1.4e-04  0.411
#> 345 1.4e-04  0.411
#> 346 1.4e-04  0.411
#> 347 1.4e-04  0.411
#> 348 1.4e-04  0.411
#> 349 1.4e-04  0.411
#> 350 1.4e-04  0.411
#> 351 1.4e-04  0.411
#> 352 1.4e-04  0.411
#> 353 1.4e-04  0.411
#> 354 1.2e-04  0.390
#> 355 1.2e-04  0.390
#> 356 1.2e-04  0.390
#> 357 1.2e-04  0.390
#> 358 1.2e-04  0.390
#> 359 1.2e-04  0.390
#> 360 1.2e-04  0.390
#> 361 1.2e-04  0.390
#> 362 1.2e-04  0.390
#> 363 1.2e-04  0.390
#> 364 1.2e-04  0.390
#> 365 1.2e-04  0.390
#> 366 1.2e-04  0.390
#> 367 1.2e-04  0.390
#> 368 1.2e-04  0.390
#> 369 1.1e-04  0.370
#> 370 1.1e-04  0.370
#> 371 1.1e-04  0.370
#> 372 1.1e-04  0.370
#> 373 1.1e-04  0.370
#> 374 1.1e-04  0.370
#> 375 1.1e-04  0.370
#> 376 1.1e-04  0.370
#> 377 1.1e-04  0.370
#> 378 1.1e-04  0.370
#> 379 1.1e-04  0.370
#> 380 1.1e-04  0.370
#> 381 1.1e-04  0.370
#> 382 1.1e-04  0.370
#> 383 1.1e-04  0.370
#> 384 1.0e-04  0.350
#> 385 1.0e-04  0.350
#> 386 1.0e-04  0.350
#> 387 1.0e-04  0.350
#> 388 1.0e-04  0.350
#> 389 1.0e-04  0.350
#> 390 1.0e-04  0.350
#> 391 1.0e-04  0.350
#> 392 1.0e-04  0.350
#> 393 9.3e-05  0.332
#> 394 9.3e-05  0.332
#> 395 9.3e-05  0.332
#> 396 9.3e-05  0.332
#> 397 9.3e-05  0.332
#> 398 9.3e-05  0.332
#> 399 9.3e-05  0.332
#> 400 9.3e-05  0.332
#> 401 9.3e-05  0.332
#> 402 9.3e-05  0.332
#> 403 9.3e-05  0.332
#> 404 9.3e-05  0.332
#> 405 9.3e-05  0.332
#> 406 8.3e-05  0.314
#> 407 8.3e-05  0.314
#> 408 8.3e-05  0.314
#> 409 8.3e-05  0.314
#> 410 8.3e-05  0.314
#> 411 8.3e-05  0.314
#> 412 8.3e-05  0.314
#> 413 8.3e-05  0.314
#> 414 8.3e-05  0.314
#> 415 8.3e-05  0.314
#> 416 8.3e-05  0.314
#> 417 8.3e-05  0.314
#> 418 7.4e-05  0.298
#> 419 7.4e-05  0.298
#> 420 7.4e-05  0.298
#> 421 7.4e-05  0.298
#> 422 7.4e-05  0.298
#> 423 7.4e-05  0.298
#> 424 7.4e-05  0.298
#> 425 7.4e-05  0.298
#> 426 7.4e-05  0.298
#> 427 7.4e-05  0.298
#> 428 7.4e-05  0.298
#> 429 7.4e-05  0.298
#> 430 7.4e-05  0.298
#> 431 7.4e-05  0.298
#> 432 7.4e-05  0.298
#> 433 7.4e-05  0.298
#> 434 6.6e-05  0.282
#> 435 6.6e-05  0.282
#> 436 6.6e-05  0.282
#> 437 6.6e-05  0.282
#> 438 6.6e-05  0.282
#> 439 6.6e-05  0.282
#> 440 6.6e-05  0.282
#> 441 6.6e-05  0.282
#> 442 6.6e-05  0.282
#> 443 6.6e-05  0.282
#> 444 6.6e-05  0.282
#> 445 6.6e-05  0.282
#> 446 6.6e-05  0.282
#> 447 6.6e-05  0.282
#> 448 5.8e-05  0.267
#> 449 5.8e-05  0.267
#> 450 5.8e-05  0.267
#> 451 5.8e-05  0.267
#> 452 5.8e-05  0.267
#> 453 5.8e-05  0.267
#> 454 5.8e-05  0.267
#> 455 5.8e-05  0.267
#> 456 5.8e-05  0.267
#> 457 5.8e-05  0.267
#> 458 5.1e-05  0.253
#> 459 5.1e-05  0.253
#> 460 5.1e-05  0.253
#> 461 5.1e-05  0.253
#> 462 5.1e-05  0.253
#> 463 5.1e-05  0.253
#> 464 5.1e-05  0.253
#> 465 5.1e-05  0.253
#> 466 5.1e-05  0.253
#> 467 4.5e-05  0.239
#> 468 4.5e-05  0.239
#> 469 4.5e-05  0.239
#> 470 4.5e-05  0.239
#> 471 4.5e-05  0.239
#> 472 4.0e-05  0.227
#> 473 4.0e-05  0.227
#> 474 4.0e-05  0.227
#> 475 4.0e-05  0.227
#> 476 4.0e-05  0.227
#> 477 4.0e-05  0.227
#> 478 4.0e-05  0.227
#> 479 4.0e-05  0.227
#> 480 4.0e-05  0.227
#> 481 3.4e-05  0.214
#> 482 3.4e-05  0.214
#> 483 3.4e-05  0.214
#> 484 3.4e-05  0.214
#> 485 3.4e-05  0.214
#> 486 3.4e-05  0.214
#> 487 3.4e-05  0.214
#> 488 3.0e-05  0.203
#> 489 3.0e-05  0.203
#> 490 2.6e-05  0.192
#> 491 2.6e-05  0.192
#> 492 2.6e-05  0.192
#> 493 2.6e-05  0.192
#> 494 2.2e-05  0.182
#> 495 2.2e-05  0.182
#> 496 2.2e-05  0.182
#> 497 2.2e-05  0.182
#> 498 2.2e-05  0.182
#> 499 1.9e-05  0.172
#> 500 1.9e-05  0.172
#> 501 1.9e-05  0.172
#> 502 1.9e-05  0.172
#> 503 1.7e-05  0.163
#> 504 1.7e-05  0.163
#> 505 1.0e-05  0.138
#> 506 7.5e-06  0.124
#> 507 5.4e-06  0.111
#> 508 3.9e-06  0.099
#> 509 3.3e-06  0.094
#> 510 2.8e-06  0.089
#> 511 2.8e-06  0.089
#> 512 2.0e-06  0.079

No influential observations.

Prediction from the model

newdata <- data.frame(Receiving.Points.Won=c(20, 50), won=c(NA, NA))
predict(tennis_glm, newdata, type="response")
#>    1    2 
#> 0.34 0.93

Interpret the response as the probability of winning if your receiving points was 20, 50.

Summary

Generalised linear models are a systematic way to fit different types of response distributions.

Resources

• Beginners guide
• Introduction to GLMs
• Quick-R GLMs
• The Analysis Factor, Generalized Linear Models Parts 1-4
• wikipedia
• Do Smashes Win Matches?