mages' blog

Visualising the predictive distribution of a log-transformed linear model

Last week I presented visualisations of theoretical distributions that predict ice cream sales statistics based on linear and generalised linear models, which I introduced in an earlier post.
 Theoretical distributions
Today I will take a closer look at the log-transformed linear model and use Stan/rstan, not only to model the sales statistics, but also to generate samples from the posterior predictive distribution.

The posterior predictive distribution is what I am most interested in. From the simulations I can get the 95% prediction interval, which will be slightly wider than the theoretical 95% interval, as it takes into account the parameter uncertainty as well.

Ok, first I take my log-transformed linear model of my earlier post and turn it into a Stan model, including a section to generate output from the posterior predictive distribution.

After I have complied and run the model, I can extract the simulations and calculate various summary statistics. Furthermore, I use my parameters also to predict the median and mean, so that I can compare them against the sample statistics. Note again, that for the mean calculation of the log-normal distribution I have to take into account the variance as well.

Ok, that looks pretty reasonable, and also quite similar to my earlier output with glm. Using my plotting function of last week I can also create a nice 3D plot again.

 Posterior predictive distributions

Just as expected, I note a slightly wider 95% interval range in the posterior predictive distributions compared to the theoretical distributions at the top.

Session Info

R version 3.2.2 (2015-08-14)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.10.5 (Yosemite)

locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils datasets
[6] methods   base

other attached packages:
[1] rstan_2.7.0-1 inline_0.3.14 Rcpp_0.12.0

loaded via a namespace (and not attached):
[1] tools_3.2.2 codetools_0.2-14 stats4_3.2.2

Visualising theoretical distributions of GLMs

Two weeks ago I discussed various linear and generalised linear models in R using ice cream sales statistics. The data showed not surprisingly that more ice cream was sold at higher temperatures.
icecream <- data.frame(
temp=c(11.9, 14.2, 15.2, 16.4, 17.2, 18.1,
18.5, 19.4, 22.1, 22.6, 23.4, 25.1),
units=c(185L, 215L, 332L, 325L, 408L, 421L,
406L, 412L, 522L, 445L, 544L, 614L)
)
I used a linear model, a log-transformed linear model, a Poisson and Binomial generalised linear model to predict sales within and outside the range of data available.

I came to the conclusion that I preferred the binomial model for two reasons: the model simulates only whole numbers, just like the observational data and it has natural boundaries defined by zero sales and a market saturation level. Hence, it will neither predict negative sales, nor infinite sales.

However, I didn't review any residual plots or anything else but the mean predictions. So, let's do this in this post.

Arthur Charpentier reminded me via Twitter of an earlier post of his, where he looked at GLMs as well. Arthur used a nice 3D plot to visualise the models. I took his code and wrapped it into a function (it is not pretty, but it will do for now):
glmModelPlot <- function(x, y, xlim,ylim, meanPred,  LwPred, UpPred,
plotData, main=NULL){
## Based on code by Arthur Charpentier:
## http://freakonometrics.hypotheses.org/9593
par(mfrow=c(1,1))
n <- 2
N <- length(meanPred)
zMax <- max(unlist(sapply(plotData, "[[", "z")))*1.5
mat <- persp(xlim, ylim, matrix(0, n, n), main=main,
zlim=c(0, zMax), theta=-30,
ticktype="detailed",box=FALSE)
C <- trans3d(x, UpPred, rep(0, N),mat)
lines(C, lty=2)
C <- trans3d(x, LwPred, rep(0, N), mat)
lines(C, lty=2)
C <- trans3d(c(x, rev(x)), c(UpPred, rev(LwPred)),
rep(0, 2*N), mat)
polygon(C, border=NA, col=adjustcolor("yellow", alpha.f = 0.5))
C <- trans3d(x, meanPred, rep(0, N), mat)
lines(C, lwd=2, col="grey")
C <- trans3d(x, y, rep(0,N), mat)
points(C, lwd=2, col="#00526D")
for(j in N:1){
xp <- plotData[[j]]$x yp <- plotData[[j]]$y
z0 <- plotData[[j]]$z0 zp <- plotData[[j]]$z
C <- trans3d(c(xp, xp), c(yp, rev(yp)), c(zp, z0), mat)
polygon(C, border=NA, col="light blue", density=40)
C <- trans3d(xp, yp, z0, mat)
lines(C, lty=2)
C <- trans3d(xp, yp, zp, mat)
}
}

Linear model

I start by plotting the output of the linear model, a GLM assuming a Normal distribution with an identity link function.

The 3D plot shows the observations as circles on the xy-plane, together with the mean predictions as a solid line. In yellow I show the theoretical residual interval between the 5th and 95th quantile of the Normal distributions parametrised by the model output. Additionally I present the density distributions of the model at each observation.

This is conceptually similar to a Q-Q, which compares the standardised residuals with the theoretical quantiles. In the case of a linear model the prediction interval would shrink to the theoretical interval as the number of data points increases.

The 3D plot looks pretty, but I think it is helpful as well. Let me put this in context to the traditional residual plots, see below.

The residual plots identify observations 2, 5 and 10 as extreme points. Indeed, those points are either on the edge or outside (point 10) of the yellow area in the plot above.

xlim <- c(min(icecream$temp)*0.95, max(icecream$temp)*1.05)
ylim <- c(floor(min(icecream$units)*0.95), ceiling(max(icecream$units)*1.05))

lin.mod <- glm(units ~ temp, data=icecream,
par(mfrow=c(2,2))
plot(lin.mod)
title(outer=TRUE, line = -1,
main = list("Linear regression",
cex=1.25,col="black", font=2))

meanPred <- predict(lin.mod, type="response")
sdgig <- sqrt(summary(lin.mod)$dispersion) UpPred <- qnorm(.95, meanPred, sdgig) LwPred <- qnorm(.05, meanPred, sdgig) plotData <- lapply( seq(along=icecream$temp),
function(i){
stp <- 251
x = rep(icecream$temp[i], stp) y = seq(ylim[1], ylim[2], length=stp) z0 = rep(0, stp) z = dnorm(y, meanPred[i], sdgig) return(list(x=x, y=y, z0=z0, z=z)) } ) glmModelPlot(x = icecream$temp, y=icecream$units, xlim=xlim, ylim=ylim, meanPred = meanPred, LwPred = LwPred, UpPred = UpPred, plotData = plotData, main = "Linear regression") Log-transformed linear model The log-transformed model shows an increasing variance around the mean as the temperatures increase. Yet, this shouldn't come as a surprise, as this a property of the log-normal distribution. The variance is given as $$\text{Var}(X) = \exp(2\mu + \sigma^2)(\exp(\sigma^2) - 1) = \text{E}[X]^2 (\exp(\sigma^2) - 1)$$ . Again, the residual plots highlight points 2, 5 and 10 as extremes. I note also from the first residual plot that the log-transformed model over-predicts for colder temperatures. log.lin.mod <- glm(log(units) ~ temp, data=icecream, family=gaussian(link="identity")) par(mfrow=c(2,2)) plot(log.lin.mod) title(outer=TRUE, line = -1, main = list("Log-transformed LM", cex=1.25,col="black", font=2)) meanLogPred <- predict(log.lin.mod, type="response") sdgig <- sqrt(summary(log.lin.mod)$dispersion)

meanPred <- exp(meanLogPred + 0.5 * sdgig^2)
UpPred <- qlnorm(.95, meanlog = meanLogPred, sdlog = sdgig)
LwPred <- qlnorm(.05, meanlog = meanLogPred, sdlog = sdgig)

plotData <- lapply(
seq(along=icecream$temp), function(i){ stp <- 251 x = rep(icecream$temp[i], stp)
y = seq(ylim[1], ylim[2], length=stp)
z0 = rep(0, stp)
z = dlnorm(y, meanlog = meanLogPred[i], sdlog = sdgig)
return(list(x=x, y=y, z0=z0, z=z))
}
)

glmModelPlot(x = icecream$temp, y=icecream$units,
xlim=xlim, ylim=ylim,
meanPred = meanPred, LwPred = LwPred,
UpPred = UpPred, plotData = plotData,
main = "Log-transformed LM")

Poisson (log) GLM

The Poisson model shows a narrower range between the 5th and 95th quantile then the previous models. My Poisson model didn't assume over-dispersion, and hence the yellow range increases in line with the mean.

The same points 2, 5 and 10 are highlighted again as extreme, but now they are well outside the yellow area.

pois.mod <- glm(units ~ temp, data=icecream,
par(mfrow=c(2,2))
plot(pois.mod)
title(outer=TRUE, line = -1,
main = list("Poisson (log) GLM",
cex=1.25,col="black", font=2))

meanPred <- predict(pois.mod, type="response")
UpPred <- qpois(.95, meanPred)
LwPred <- qpois(.05, meanPred)

plotData <- lapply(
seq(along=icecream$temp), function(i){ y = seq(ylim[1], ylim[2]) x = rep(icecream$temp[i], length(y))
z0 = rep(0, length(y))
z = dpois(y, meanPred[i])
return(list(x=x, y=y, z0=z0, z=z))
}
)

glmModelPlot(x = icecream$temp, y=icecream$units,
xlim=xlim, ylim=ylim,
meanPred = meanPred, LwPred = LwPred,
UpPred = UpPred, plotData = plotData,
main = "Poisson (log) GLM")

Binomial (logit) GLM

Here is my favourite model from my previous post again.

The theoretical range between the 5th and 95th quantile has narrowed further. Data item 10 looks now really suspicious. Observations 2 and 5 are once again highlighted. Now I wonder, is someone eating my ice cream?

market.size <- 800
icecream$opportunity <- market.size - icecream$units
bin.glm <- glm(cbind(units, opportunity) ~ temp, data=icecream,
par(mfrow=c(2,2))
plot(bin.glm)
title(outer=TRUE, line = -1,
main = list("Binomial (logit) GLM",
cex=1.25,col="black", font=2))

meanProb <- predict(bin.glm, type="response")
meanPred <- meanProb*market.size
UpPred <- qbinom(.95, market.size, meanProb)
LwPred <- qbinom(.05, market.size, meanProb)

plotData <- lapply(
seq(along=icecream$temp), function(i){ y = ylim[1]:ylim[2] x = rep(icecream$temp[i], length(y))
z0 = rep(0, length(y))
z = dbinom(y, market.size, meanProb[i])
return(list(x=x, y=y, z0=z0, z=z))
}
)

glmModelPlot(x = icecream$temp, y=icecream$units,
xlim=xlim, ylim=ylim,
meanPred = meanPred, LwPred = LwPred,
UpPred = UpPred, plotData = plotData,
main = "Binomial (logit) GLM")

Conclusions

I think Arthur's 3D plots help to visualise what GLMs are conceptually about. They illustrate the theoretical distribution around the predictions. Let me say this again, the yellow areas in the charts above show not the 90% prediction interval, but the theoretical residual distribution if the models where true.

Session Info

R version 3.2.2 (2015-08-14)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.10.4 (Yosemite)

locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8

attached base packages:
[1] stats graphics grDevices utils datasets methods base

loaded via a namespace (and not attached):
[1] tools_3.2.2 Rcpp_0.12.0

Generalised Linear Models in R

Linear models are the bread and butter of statistics, but there is a lot more to it than taking a ruler and drawing a line through a couple of points.

Some time ago Rasmus Bååth published an insightful blog article about how such models could be described from a distribution centric point of view, instead of the classic error terms convention.

I think the distribution centric view makes generalised linear models (GLM) much easier to understand as well. That’s the purpose of this post.

Using data on ice cream sales statistics I will set out to illustrate different models, starting with traditional linear least square regression, moving on to a linear model, a log-transformed linear model and then on to generalised linear models, namely a Poisson (log) GLM and Binomial (logistic) GLM. Additionally, I will run a simulation with each model.

Along the way I aim to reveal the intuition behind a GLM using Ramus’ distribution centric description. I hope to clarify why one would want to use different distributions and link functions.

The data

Here is the example data set I will be using. It shows the number of ice creams sold at different temperatures.
icecream <- data.frame(
temp=c(11.9, 14.2, 15.2, 16.4, 17.2, 18.1,
18.5, 19.4, 22.1, 22.6, 23.4, 25.1),
units=c(185L, 215L, 332L, 325L, 408L, 421L,
406L, 412L, 522L, 445L, 544L, 614L)
)
I will be comparing the results from various models and hence, I write a helper function to provide consistent graphical outputs:
basicPlot <- function(...){
plot(units ~ temp, data=icecream, bty="n", lwd=2,
main="Number of ice creams sold", col="#00526D",
xlab="Temperature (Celsius)",
ylab="Units sold", ...)
axis(side = 1, col="grey")
axis(side = 2, col="grey")
}
basicPlot()
As expected more ice cream is sold at higher temperature.

The challenge

I would like to create a model that predicts the number of ice creams sold for different temperatures, even outside the range of the available data.

I am particularly interested in how my models will behave in the more extreme cases when it is freezing outside, say when the temperature drops to 0ºC and also what it would say for a very hot summer's day at 35ºC.

Linear least square

My first approach is to take a ruler and draw a straight line through the points, such that it minimises the average distance between the points and the line. That's basically a linear least square regression line:
lsq.mod <- lsfit(icecream$temp, icecream$units)
basicPlot()
abline(lsq.mod, col="orange", lwd=2)
legend(x="topleft", bty="n", lwd=c(2,2), lty=c(NA,1),
legend=c("observation", "linear least square"),
col=c("#00526D","orange"),  pch=c(1,NA))
That's easy and looks not unreasonable.

Linear regression

For the least square method I didn't think about distributions at all. It is just common sense, or is it?

I believe the observation $$y_i$$ was drawn from a Normal (aka Gaussian) distribution with a mean $$\mu_i$$, depending on the temperature $$x_i$$ and a constant variance $$\sigma^2$$ across all temperatures.

On another day with the same temperature I might have sold a different quantity of ice cream, but over many days with the same temperature the number of ice creams sold would start to fall into a range defined by $$\mu_i\pm\sigma$$.

Thus, using a distribution centric notation my model looks like this:
$y_i \sim \mathcal{N}(\mu_i, \sigma^2), \\ \mathbb{E}[y_i]=\mu_i = \alpha + \beta x_i \; \text{for all}\; i$Note, going forward I will drop the $$\text{for all}\; i$$ statement for brevity.

Or, alternatively I might argue that the residuals, the difference between the observations and predicted values, follow a Gaussian distribution with a mean of 0 and variance $$\sigma^2$$:
$\varepsilon_i = y_i - \mu_i \sim \mathcal{N}(0, \sigma^2)$Furthermore, the equation
$\mathbb{E}[y_i]=\mu_i = \alpha + \beta x_i$states my belief that the expected value of $$y_i$$ is identical to the parameter $$\mu_i$$ of the underlying distribution, while the variance is constant.

The same model written in the classic error terms convention would be:
$y_i = \alpha + \beta x_i + \varepsilon_i, \text{ with } \varepsilon_i \sim \mathcal{N}(0, \sigma^2)$I think the probability distribution centric convention makes it clearer that my observation is just one realisation from a distribution. It also emphasises that the parameter of the distribution is modelled linearly.

To model this in R explicitly I use the glm function, in which I specify the "response distribution" (namely the number of ice creams) as Gaussian and the link function from the expected value of the distribution to its parameter (i.e. temperature) as identity. Indeed, this really is the trick with a GLM, it describes how the distribution of the observations and its expected value, often after a smooth transformation, relate to the actual measurements (predictors) in a linear way. The 'link' is the inverse function of the original transformation of the data.

That's what its says on the GLM tin and that's all there is to it!

(Actually, there is also the bit, which estimates the parameters from the data via maximum likelihood, but I will skip this here.)

lin.mod <- glm(units ~ temp, data=icecream,
library(arm) # for 'display' function only
display(lin.mod)
## glm(formula = units ~ temp, family = gaussian(link = "identity"),
##     data = icecream)
##             coef.est coef.se
## (Intercept) -159.47    54.64
## temp          30.09     2.87
## ---
##   n = 12, k = 2
##   residual deviance = 14536.3, null deviance = 174754.9 (difference = 160218.6)
##   overdispersion parameter = 1453.6
##   residual sd is sqrt(overdispersion) = 38.13
Thus, to mimic my data I could generate random numbers from the following Normal distribution:
$y_i \sim \mathcal{N}(\mu_i, \sigma^2) \text{ with } \mu_i = -159.5 + 30.1 \; x_i \text{ and } \sigma = 38.1$Although the linear model looks fine in the range of temperatures observed, it doesn't make much sense at 0ºC. The intercept is at -159, which would mean that customers return on average 159 units of ice cream on a freezing day. Well, I don't think so.

Log-transformed linear regression

Ok, perhaps I can transform the data first. Ideally I would like ensure that the transformed data has only positive values. The first transformation that comes to my mind in those cases is the logarithm.

So, let's model the ice cream sales on a logarithmic scale. Thus my model changes to:
$\log(y_i) \sim \mathcal{N}(\mu_i, \sigma^2)\\ \mathbb{E}[\log(y_i)]=\mu_i = \alpha + \beta x_i$This model implies that I believe the sales follow a log-normal distribution:$y_i \sim \log\mathcal{N}(\mu_i, \sigma^2)$Note, the log-normal distribution is skewed to the right, which means that I regard higher sales figures more likely than lower sales figures.

Although the model is still linear on a log-scale, I have to remember to transform the predictions back to the original scale because $$\mathbb{E}[\log(y_i)] \neq \log(\mathbb{E}[y_i])$$. This is shown below. For a discussion of the transformation of the lognormal distribution see for example the help page of the R function rlnorm.
$y_i \sim \log\mathcal{N}(\mu_i, \sigma^2)\\ \mathbb{E}[y_i] = \exp(\mu_i + \sigma^2/2) = \exp(\alpha + \beta x_i + \sigma^2/2)$
log.lin.mod <- glm(log(units) ~ temp, data=icecream,
display(log.lin.mod)
## glm(formula = log(units) ~ temp, family = gaussian(link = "identity"),
##     data = icecream)
##             coef.est coef.se
## (Intercept) 4.40     0.20
## temp        0.08     0.01
## ---
##   n = 12, k = 2
##   residual deviance = 0.2, null deviance = 1.4 (difference = 1.2)
##   overdispersion parameter = 0.0
##   residual sd is sqrt(overdispersion) = 0.14
log.lin.sig <- summary(log.lin.mod)$dispersion log.lin.pred <- exp(predict(log.lin.mod) + 0.5*log.lin.sig) basicPlot() lines(icecream$temp, log.lin.pred, col="red", lwd=2)
legend(x="topleft", bty="n", lwd=c(2,2), lty=c(NA,1),
legend=c("observation", "log-transformed LM"),
col=c("#00526D","red"), pch=c(1,NA))


This plot looks a little better than the previous linear model and it predicts that I would sell, on average, 82 ice creams when the temperature is 0ºC:
exp(coef(log.lin.mod)[1] + 0.5 * log.lin.sig)
## (Intercept)
##    82.3945
Although this model makes a little more sense, it appears that is predicts too many sales at the low and high-end of the observed temperature range. Furthermore, there is another problem with this model and the previous linear model as well. The assumed model distributions generate real numbers, but ice cream sales only occur in whole numbers. As a result, any draw from the model distribution should be a whole number.

Poisson regression

The classic approach for count data is the Poisson distribution.

The Poisson distribution has only one parameter, here $$\mu_i$$, which is also its expected value. The canonical link function for $$\mu_i$$ is the logarithm, i.e. the logarithm of the expected value is regarded a linear function of the predictors. This means I have to apply the exponential function to the linear model to get back to the original scale. This is distinctively different to the log-transformed linear model above, where the original data was transformed, not the expected value of the data. The log function here is the link function, as it links the transformed expected value to the linear model.

Here is my model:
$y_i \sim \text{Poisson}(\mu_i)\\ \mathbb{E}[y_i]=\mu_i=\exp(\alpha + \beta x_i) = \exp(\alpha) \exp(\beta x_i)\\ \log(\mu_i) = \alpha + \beta x_i$Let me say this again, although the expected value of my observation is a real number, the Poisson distribution will generate only whole numbers, in line with the actual sales.
pois.mod <- glm(units ~ temp, data=icecream,
display(pois.mod)
## glm(formula = units ~ temp, family = poisson(link = "log"), data = icecream)
##             coef.est coef.se
## (Intercept) 4.54     0.08
## temp        0.08     0.00
## ---
##   n = 12, k = 2
##   residual deviance = 60.0, null deviance = 460.1 (difference = 400.1)
pois.pred <- predict(pois.mod, type="response")
basicPlot()
lines(icecream$temp, pois.pred, col="blue", lwd=2) legend(x="topleft", bty="n", lwd=c(2,2), lty=c(NA,1), legend=c("observation", "Poisson (log) GLM"), col=c("#00526D","blue"), pch=c(1,NA))  This looks pretty good. The interpretation of the coefficients should be clear now. I have to use the exp function to make a prediction of sales for a given temperature. However, R will do this for me automatically, if I set in the predict statement above type="response". From the coefficients I can read off that a 0ºC I expect to sell $$\exp(4.45)=94$$ ice creams and that with each one degree increase in temperature the sales are predicted to increase by $$\exp(0.076) - 1 = 7.9\%$$. Note, the exponential function turns the additive scale into a multiplicative one. So far, so good. My model is in line with my observations. Additionally, it will not predict negative sales and if I would simulate from a Poisson distribution with a mean given by the above model I will always only get whole numbers back. However, my model will also predict that I should expect to sell over 1000 ice creams if the temperature reaches 32ºC: predict(pois.mod, newdata=data.frame(temp=32), type="response") ## 1 ## 1056.651  Perhaps the exponential growth in my model looks a little too good to be true. Indeed, I am pretty sure that my market will be saturated at around 800. Warning: this is a modelling assumption from my side! Binomial regression Ok, let's me think about the problem this way: If I have 800 potential sales then I'd like to understand the proportion sold at a given temperature. This suggests a binomial distribution for the number of successful sales out of 800. The key parameter for the binomial distribution is the probability of success, the probability that someone will buy ice cream as a function of temperature. Dividing my sales statistics by 800 would give me a first proxy for the probability of selling all ice cream. Therefore I need an S-shape curve that maps the sales statistics into probabilities between 0 and 100%. A canonical choice is the logistic function: $\text{logit}(u) = \frac{e^u}{e^u + 1} = \frac{1}{1 + e^{-u}}$ With that my model can be described as: $y_i \sim \text{Binom}(n, \mu_i)\\ \mathbb{E}[y_i]=\mu_i=\text{logit}^{-1}(\alpha + \beta x_i)\\ \text{logit}(\mu_i) = \alpha + \beta x_i$ market.size <- 800 icecream$opportunity <- market.size - icecream$units bin.glm <- glm(cbind(units, opportunity) ~ temp, data=icecream, family=binomial(link = "logit")) display(bin.glm) ## glm(formula = cbind(units, opportunity) ~ temp, family = binomial(link = "logit"), ## data = icecream) ## coef.est coef.se ## (Intercept) -2.97 0.11 ## temp 0.16 0.01 ## --- ## n = 12, k = 2 ## residual deviance = 84.4, null deviance = 909.4 (difference = 825.0) bin.pred <- predict(bin.glm, type="response")*market.size basicPlot() lines(icecream$temp, bin.pred, col="purple", lwd=2)
legend(x="topleft", bty="n", lwd=c(2,2), lty=c(NA,1),
legend=c("observation", "Binomial (logit) GLM"),
col=c("#00526D","purple"),  pch=c(1,NA))


The chart doesn't look too bad at all!

As the temperature increases higher and higher this model will predict that sales will reach market saturation, while all the other models so far would predict higher and higher sales.

I can predict sales at 0ºC and 35ºC using the inverse of the logistic function, which is given in R as plogis:

# Sales at 0 Celsius
plogis(coef(bin.glm)[1])*market.size
## (Intercept)
##    39.09618
# Sales at 35 Celsius
plogis(coef(bin.glm)[1] +  coef(bin.glm)[2]*35)*market.size
## (Intercept)
##    745.7449
So, that is 39 ice creams at 0ºC and 746 ice creams at 35ºC. Yet, these results will of course change if I change my assumptions on the market size. A market size of 1,000 would suggest that I can sell 55 units at 0ºC and 846 at 35ºC.

Summary

Let's bring all the models together into one graph, with temperatures ranging from 0 to 35ºC.
temp <- 0:35
p.lm <- predict(lin.mod, data.frame(temp=temp), type="response")
p.log.lm <- exp(predict(log.lin.mod, data.frame(temp=0:35), type="response") +
0.5 * summary(log.lin.mod)$dispersion) p.pois <- predict(pois.mod, data.frame(temp=temp), type="response") p.bin <- predict(bin.glm, data.frame(temp=temp), type="response")*market.size basicPlot(xlim=range(temp), ylim=c(-20,market.size)) lines(temp, p.lm, type="l", col="orange", lwd=2) lines(temp, p.log.lm, type="l", col="red", lwd=2) lines(temp, p.pois, type="l", col="blue", lwd=2) lines(temp, p.bin, type="l", col="purple", lwd=2) legend(x="topleft", legend=c("observation", "linear model", "log-transformed LM", "Poisson (log) GLM", "Binomial (logit) GLM"), col=c("#00526D","orange", "red", "blue", "purple"), bty="n", lwd=rep(2,5), lty=c(NA,rep(1,4)), pch=c(1,rep(NA,4)))  The chart shows the predictions of my four models over a temperature range from 0 to 35ºC. Although the linear model looks OK between 10 and perhaps 30ºC, it shows clearly its limitations. The log-transformed linear and Poisson models appear to give similar predictions, but will predict an ever accelerating increase in sales as temperature increases. I don't believe this makes sense as even the most ice cream loving person can only eat so much ice cream on a really hot day. And that's why I would go with the Binomial model to predict my ice cream sales. Simulations Having used the distribution centric view to describe my models leads naturally to simulations. If the models are good, then I shouldn't be able to identify the simulation from the real data. In all my models the linear structure was $g(\mu_i) = \alpha + \beta x_i$or in matrix notation $g(\mu) = A v$ with $$A_{i,\cdot} = [1, x_i]$$ and $$v=[\alpha, \beta]$$, whereby $$A$$ is the model matrix, $$v$$ the vector of coefficients and $$g$$ the link function. With that being said, let's simulate data from each distribution for the temperatures measured in the original data and compare against the actual sales units. n <- nrow(icecream) A <- model.matrix(units ~ temp, data=icecream) set.seed(1234) (rand.normal <- rnorm(n, mean=A %*% coef(lin.mod), sd=sqrt(summary(lin.mod)$dispersion)))
##  [1] 152.5502 278.3509 339.2073 244.5335 374.3981 404.4103 375.2385
##  [8] 403.3892 483.9470 486.5775 526.3881 557.6662
(rand.logtrans <- rlnorm(n,
meanlog=A %*% coef(log.lin.mod),
sdlog=sqrt(summary(log.lin.mod)$dispersion))) ## [1] 196.0646 266.1002 326.8102 311.5621 315.0249 321.1920 335.3733 ## [8] 564.6961 515.9348 493.4822 530.8356 691.2354 (rand.pois <- rpois(n, lambda=exp(A %*% coef(pois.mod)))) ## [1] 220 266 279 337 364 434 349 432 517 520 516 663 (rand.bin <- rbinom(n, size=market.size, prob=plogis(A %*% coef(bin.glm)))) ## [1] 197 249 302 313 337 386 387 414 516 538 541 594 basicPlot(ylim=c(100,700)) cols <- adjustcolor(c("orange", "red", "blue", "purple"), alpha.f = 0.75) points(icecream$temp, rand.normal, pch=19, col=cols[1])
points(icecream$temp, rand.logtrans, pch=19, col=cols[2]) points(icecream$temp, rand.pois, pch=19, col=cols[3])
points(icecream\$temp, rand.bin, pch=19, col=cols[4])
legend(x="topleft",
legend=c("observation",
"linear model",
"log-transformed LM",
"Poisson (log) GLM",
"Binomial (logit) GLM"),
col=c("#00526D",cols), lty=NA,
bty="n", lwd=rep(2,5),
pch=c(1,rep(19,4)))

The chart displays one simulation per observation from each model, but I believe it shows already some interesting aspects. Not only do I see that the Poisson and Binomial model generate whole numbers, while the Normal and log-transformed Normal predict real numbers, I notice the skewness of the log-normal distribution in the red point at 19.4ºC.

Furthermore, the linear model predicts equally likely figures above and below the mean and at 16.4ºC the prediction appears a little low - perhaps as a result.

Additionally the high sales prediction at 25.1ºC of the log-transformed Normal and Poisson model shouldn't come as a surprise either.

Again, the simulation of the binomial model seems the most realistic to me.

Conclusions

I hope this little article illustrated the intuition behind generalised linear models. Fitting a model to data requires more than just applying an algorithm. In particular it is worth to think about:
• the range of expected values: are they bounded or range from $$-\infty$$ to $$\infty$$?
• the type of observations: do I expect real numbers, whole numbers or proportions?
• how to link the distribution parameters to the observations

There are many aspects of GLMs which I haven't touched on here, such as:
• all the above models incorporate a fixed level of volatility. However, in practice, the variability of making a sale at low temperatures might be significantly different than at high temperatures. Check the residual plots and consider an over-dispersed model.
• I used so called canonical link functions in my models, which have nice theoretical properties, but other choices are possible as well.
• the distribution has to be chosen from the exponential family, e.g. Normal, Gamma, Poisson, binomial, Tweedie, etc.
• GLMs use maximum likelihood as the criteria for fitting the models. I sketched the difference between least square and maximum likelihood in an earlier post.

Session Info

R version 3.2.1 (2015-06-18)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.10.4 (Yosemite)

locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8

attached base packages:
[1] stats graphics grDevices utils datasets methods base

other attached packages:
[1] arm_1.8-6 lme4_1.1-8 Matrix_1.2-1 MASS_7.3-42

loaded via a namespace (and not attached):
[1] minqa_1.2.4     tools_3.2.1     coda_0.17-1     abind_1.4-3
[5] Rcpp_0.11.6     splines_3.2.1   nlme_3.1-121    grid_3.2.1
[9] nloptr_1.0.4    lattice_0.20-31