Stats Homework (R)Part 1 of 2

• Read section 11 of handbook_statsV13• What is the difference between these two models?– g1 = glm(admit ~ ., data=mydata)– g2 = glm(admit ~ ., data=mydata, family=“binomial”)

• Compute MSE (mean squared error)– mean((g1$fitted.values – mydata$admit)^2)– mean((g2$fitted.values – mydata$admit)^2)

• Which model is better in terms of MSE

Stats Homework (R)Part 2 of 2

• Consider two loss functions– MSE.loss = function(y, yhat) mean((y - yhat)^2)– log.loss = function(y, yhat) -mean(y * log(yhat) + (1-y) * log(1-yhat))

• Which model (g1 or g2) is better in terms of loss2?– Hint: log.loss assumes yhat is a probability between 0 and 1– What are range(g1$fitted.values) and range(g2$fitted.values)?

• Let s = g1$fitted.values > 0 & g1$fitted.values < 1– What is log.loss(mydata$admit[s], g1$fitted.values[s])?

• Let baseline.fit be rep(mean(admit), length(admit))– How do the two models above compare to this baseline– Compare both g1 and g2 to this baseline using both loss functions

Functions

• fact = function(x) if(x < 2) 1 else x*fact(x-1)• fact(5)• fact2 = function(x) gamma(x+1)• fact(5)• fact(1:5)• fact(seq(1,5,0.5))

help

(sw

iss)

What are statistics packages good for?

• Plotting– Scatter plots, histograms, boxplots

• Visualization– Dendrograms, heatmaps

• Modeling– Linear Regression– Logistic Regression

• Hypothesis Testing

ObjectsData Types• Numbers: 1, 1.5• Strings• Functions:

– sqrt, help, summary• Vectors:

– 3:30– c(1:5, 25:30)– rnorm(100, 0, 1), rbinom(100, 10, .5)– rep(0,10)– scan( filename, 0)

• Matrices– matrix(rep(0,100), ncol=10)– diag(rep(1,10))– cbind(1:10, 21:30)– rbind(1:10, 21:30)

• Data Frames– swiss– read.table (filename, header=T)

Operations• All: summary, help, print, cat• Numbers: +,*,sqrt• Strings: grep, substr, paste• Vectors: x = (3:30)

– x+5– plot(x)– mean(x), var(x), median(x)– mean((x – mean(x))^2)– x[3], x[3:6], x[-1], x[x>20]

• Matrices: m = matrix(1:100,ncol=10)– m+3, m* 3, m %*% m– m[3,3], m[1:3,1:3], m[3,], m[m < 10]– diag(m), t(m)

• Data Frames– names(swiss)– plot(swiss$Education, swiss$Fertility)– plot(swiss[,1], swiss[,2])– pairs(swiss)

plot(swiss$Education, swiss$Fertility)

plot(swiss$Education, swiss$Fertility, col = 3 + (swiss$Catholic > 50))

boxplot(split(swiss$Fertility, swiss$Catholic > 50))

boxplot(split(swiss$Fertility, swiss$Education > median(swiss$Education)))

pairs(swiss)

Correlations

> cor(swiss$Fertility, swiss$Catholic)[1] 0.4636847> cor(swiss$Fertility, swiss$Education)[1] -0.6637889> round(100*cor(swiss)) Fertility Agriculture Examination Education Catholic Infant.MortalityFertility 100 35 -65 -66 46 42Agriculture 35 100 -69 -64 40 -6Examination -65 -69 100 70 -57 -11Education -66 -64 70 100 -15 -10Catholic 46 40 -57 -15 100 18Infant.Mortality 42 -6 -11 -10 18 100

• Many functions in R are polymorphic: – cor works on vectors as well as matrices

Regression> plot(swiss$Education, swiss$Fertility)> g = glm(swiss$Fertility ~ swiss$Education)> abline(g)> summary(g)

Call:glm(formula = swiss$Fertility ~ swiss$Education)

Deviance Residuals: Min 1Q Median 3Q Max -17.036 -6.711 -1.011 9.526 19.689

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 79.6101 2.1041 37.836 < 2e-16 ***swiss$Education -0.8624 0.1448 -5.954 3.66e-07 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 89.22746)

Null deviance: 7178.0 on 46 degrees of freedomResidual deviance: 4015.2 on 45 degrees of freedomAIC: 348.42

Number of Fisher Scoring iterations: 2

plot(swiss$Education, swiss$Fertility, type='n’); abline(g)text(swiss$Education, swiss$Fertility, dimnames(swiss)[[1]])

g = glm(Fertility ~ . , data=swiss); summary(g)

Call:glm(formula = Fertility ~ ., data = swiss)

Deviance Residuals: Min 1Q Median 3Q Max -15.2743 -5.2617 0.5032 4.1198 15.3213

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 66.91518 10.70604 6.250 1.91e-07 ***Agriculture -0.17211 0.07030 -2.448 0.01873 * Examination -0.25801 0.25388 -1.016 0.31546 Education -0.87094 0.18303 -4.758 2.43e-05 ***Catholic 0.10412 0.03526 2.953 0.00519 ** Infant.Mortality 1.07705 0.38172 2.822 0.00734 ** ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 51.34251)

Null deviance: 7178.0 on 46 degrees of freedomResidual deviance: 2105.0 on 41 degrees of freedomAIC: 326.07

Number of Fisher Scoring iterations: 2

http://www.nytimes.com/2009/01/07/technology/business-computing/07program.html

Plot Example

• help(ldeaths)• plot(mdeaths, col="blue", ylab="Deaths", sub="Male

(blue), Female (pink)", ylim=range(c(mdeaths, fdeaths)))• lines(fdeaths, lwd=3, col="pink")• abline(v=1970:1980, lty=3)• abline(h=seq(0,3000,1000), lty=3, col="red")

Periodicity

• plot(1:length(fdeaths), fdeaths, type='l')• lines((1:length(fdeaths))+12, fdeaths, lty=3)

Type Argument

par(mfrow=c(2,2))plot(fdeaths, type='p', main="points")plot(fdeaths, type='l', main="lines")plot(fdeaths, type='b', main="b")plot(fdeaths, type='o', main="o")

ScatterPlot

• plot(as.vector(mdeaths), as.vector(fdeaths))• g=glm(fdeaths ~ mdeaths)• abline(g)• g$coef

(Intercept) mdeaths -45.2598005 0.4050554

Hist & Density

• par(mfrow=c(2,1))• hist(fdeaths/mdeaths, nclass=30)• plot(density(fdeaths/mdeaths))

Data Frames

• help(cars)• names(cars)• summary(cars)• plot(cars)

• cars2 = cars• cars2$speed2 = cars$speed^2• cars2$speed3 = cars$speed^3• summary(cars2)• names(cars2)• plot(cars2)• options(digits=2)• cor(cars2)

Normalitypar(mfrow=c(2,1))plot(density(cars$dist/

cars$speed))lines(density(rnorm(1000000,

mean(cars$dist/cars$speed), sqrt(var(cars$dist/ cars$speed)))), col="red")

qqnorm(cars$dist/cars$speed)

abline(mean(cars$dist/cars$speed), sqrt(var(cars$dist/cars$speed)))

Stopping Distance Increases Quickly With Speed

• plot(cars$speed, cars$dist/cars$speed)

• boxplot(split(cars$dist/cars$speed, round(cars$speed/10)*10))

Quadratic Model of Stopping Distance

plot(cars$speed, cars$dist)cars$speed2 = cars$speed^2g2 = glm(cars$dist ~ cars$speed2)lines(cars$speed, g2$fitted.values)

Bowed Residuals

g1 = glm(dist ~ poly(speed, 1), data=cars)

g2 = glm(dist ~ poly(speed, 2), data=cars)

par(mfrow=c(1,2))boxplot(split(g1$resid,

round(cars$speed/5))); abline(h=0)

boxplot(split(g2$resid, round(cars$speed/5))); abline(h=0)

Help, Demo, Example

• demo(graphics)– example(plot)– example(lines)– help(cars)– help(WWWusage)– example(abline)– example(text)– example(par)

• boxplots– example(boxplot)– help(chickwts)

• demo(plotmath)

• pairs– example(pairs)– help(quakes)– help(airquality)– help(attitude)– Anorexia

• utils::data(anorexia, package="MASS")

• pairs(anorexia, col=c("red", "green", "blue")[anorexia$Treat])

• counting– example(table)– example(quantile)– example(hist)– help(faithful)

• Randomness– example(rnorm)– example(rbinom)– example(rt)

• Normality– example(qqnorm)

• Regression– help(cars)– example(glm)– demo(lm.glm)