Now we are going to cover how to perform a variety of basic statistical tests in R.

• Correlation
• T-tests
• Linear Regression
• Logistic Regression
• Proportion tests
• Chi-squared
• Fisher's Exact Test

Note: We will be glossing over the statistical theory and "formulas" for these tests. There are plenty of resources online for learning more about these tests, as well as dedicated Biostatistics series at the School of Public Health

For many of these testing result objects, you can extract specific slots/results as numbers, as the ct object is just a list.

> # str(ct)
> names(ct)

[1] "statistic"   "parameter"   "p.value"     "estimate"    "null.value" 
[6] "alternative" "method"      "data.name"   "conf.int"   

> ct$statistic

       t 
73.65553 

> ct$p.value

[1] 0

The T-test is performed using the t.test() function, which essentially tests for the difference in means of a variable between two groups.

In this syntax, x and y are the column of data for each group.

> tt = t.test(circ$orangeAverage, circ$purpleAverage)
> tt

    Welch Two Sample t-test

data:  circ$orangeAverage and circ$purpleAverage
t = -17.076, df = 1984, p-value < 2.2e-16
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -1096.7602  -870.7867
sample estimates:
mean of x mean of y 
 3033.161  4016.935 

t.test saves a lot of information: the difference in means estimate, confidence interval for the difference conf.int, the p-value p.value, etc.

> names(tt)

[1] "statistic"   "parameter"   "p.value"     "conf.int"    "estimate"   
[6] "null.value"  "alternative" "method"      "data.name"  

You can also use the 'formula' notation. In this syntax, it is y ~ x, where x is a factor with 2 levels or a binary variable and y is a vector of the same length.

> http_data_dir = "http://www.aejaffe.com/summerR_2016/data/"
> cars = read.csv(paste0(http_data_dir, "kaggleCarAuction.csv"),
+                 as.is=TRUE)
> tt2 = t.test(VehBCost~IsBadBuy, data=cars)
> tt2$estimate

mean in group 0 mean in group 1 
       6797.077        6259.274 

You can add the t-statistic and p-value to a boxplot.

> boxplot(VehBCost~IsBadBuy, data=cars, 
+         xlab="Bad Buy",ylab="Value")
> leg = paste("t=", signif(tt$statistic,3), 
+     " (p=",signif(tt$p.value,3),")",sep="")
> legend("topleft", leg, cex=1.2, bty="n")

Now we will briefly cover linear regression. I will use a little notation here so some of the commands are easier to put in the proper context. \[ y_i = \alpha + \beta x_i + \varepsilon_i \] where:

• \(y_i\) is the outcome for person i
• \(\alpha\) is the intercept
• \(\beta\) is the slope
• \(x_i\) is the predictor for person i
• \(\varepsilon_i\) is the residual variation for person i

The R version of the regression model is:

y ~ x

where:

• y is your outcome
• x is/are your predictor(s)

For a linear regression, when the predictor is binary this is the same as a t-test:

> fit = lm(VehBCost~IsBadBuy, data=cars)
> fit

Call:
lm(formula = VehBCost ~ IsBadBuy, data = cars)

Coefficients:
(Intercept)     IsBadBuy  
     6797.1       -537.8  

'(Intercept)' is \(\alpha\)

'IsBadBuy' is \(\beta\)

The summary command gets all the additional information (p-values, t-statistics, r-square) that you usually want from a regression.

> sfit = summary(fit)
> print(sfit)

Call:
lm(formula = VehBCost ~ IsBadBuy, data = cars)

Residuals:
   Min     1Q Median     3Q    Max 
 -6258  -1297    -27   1153  39210 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 6797.077      6.953  977.61   <2e-16 ***
IsBadBuy    -537.803     19.826  -27.13   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1759 on 72981 degrees of freedom
Multiple R-squared:  0.009982,  Adjusted R-squared:  0.009969 
F-statistic: 735.9 on 1 and 72981 DF,  p-value: < 2.2e-16

The coefficients from a summary are the coefficients, standard errors, t-statistcs, and p-values for all the estimates.

> names(sfit)

 [1] "call"          "terms"         "residuals"     "coefficients" 
 [5] "aliased"       "sigma"         "df"            "r.squared"    
 [9] "adj.r.squared" "fstatistic"    "cov.unscaled" 

> sfit$coef

             Estimate Std. Error   t value     Pr(>|t|)
(Intercept) 6797.0774   6.952728 977.61299  0.00000e+00
IsBadBuy    -537.8033  19.825525 -27.12681 3.01661e-161

We'll look at vehicle odometer value by vehicle age:

fit = lm(VehOdo~VehicleAge, data=cars)
print(fit)

## 
## Call:
## lm(formula = VehOdo ~ VehicleAge, data = cars)
## 
## Coefficients:
## (Intercept)   VehicleAge  
##       60127         2723

> library(scales) # we need this for the alpha command - make points transparent
> par(mfrow=c(1,2))
> plot(VehOdo ~ jitter(VehicleAge,amount=0.2), data=cars, pch = 19,
+      col = alpha("black",0.05), xlab="Vehicle Age (Yrs)")
> abline(fit, col="red",lwd=2)
> legend("topleft", paste("p =",summary(fit)$coef[2,4]))
> boxplot(VehOdo ~ VehicleAge, data=cars, varwidth=TRUE)
> abline(fit, col="red",lwd=2)

Note that you can have more than 1 predictor in regression models.The interpretation for each slope is change in the predictor corresponding to a one-unit change in the outcome, holding all other predictors constant.

> fit2 = lm(VehOdo ~ IsBadBuy + VehicleAge, data=cars)
> summary(fit2)  

Call:
lm(formula = VehOdo ~ IsBadBuy + VehicleAge, data = cars)

Residuals:
   Min     1Q Median     3Q    Max 
-70856  -9490   1390  10311  41193 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 60141.77     134.75  446.33   <2e-16 ***
IsBadBuy     1329.00     157.84    8.42   <2e-16 ***
VehicleAge   2680.33      30.27   88.53   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 13810 on 72980 degrees of freedom
Multiple R-squared:  0.1031,    Adjusted R-squared:  0.1031 
F-statistic:  4196 on 2 and 72980 DF,  p-value: < 2.2e-16

Added-Variable plots can show you the relationship between a variable and outcome after adjusting for other variables. The function avPlots from the car package can do this:

> library(car)
> avPlots(fit2)

Plot on an lm object will do diagnostic plots. Residuals vs. Fitted should have no discernable shape (the red line is the smoother), the qqplot shows how well the residuals fit a normal distribution, and Cook's distance measures the influence of individual points.

> par(mfrow=c(2,2))
> plot(fit2, ask= FALSE)

Note the coefficients are on the original scale, we must exponentiate them for odds ratios:

> exp(coef(glmfit))

(Intercept)      VehOdo  VehicleAge 
 0.02286316  1.00000834  1.30748911 

prop.test() can be used for testing the null that the proportions (probabilities of success) in several groups are the same, or that they equal certain given values.

prop.test(x, n, p = NULL,
          alternative = c("two.sided", "less", "greater"),
          conf.level = 0.95, correct = TRUE)

> prop.test(x=15, n =32)

    1-sample proportions test with continuity correction

data:  15 out of 32, null probability 0.5
X-squared = 0.03125, df = 1, p-value = 0.8597
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
 0.2951014 0.6496695
sample estimates:
      p 
0.46875 

chisq.test() performs chi-squared contingency table tests and goodness-of-fit tests.

chisq.test(x, y = NULL, correct = TRUE,
           p = rep(1/length(x), length(x)), rescale.p = FALSE,
           simulate.p.value = FALSE, B = 2000)

> tab = table(cars$IsBadBuy, cars$IsOnlineSale)
> tab

   
        0     1
  0 62375  1632
  1  8763   213

You can also pass in a table object (such as tab here)

> cq=chisq.test(tab)
> cq

    Pearson's Chi-squared test with Yates' continuity correction

data:  tab
X-squared = 0.92735, df = 1, p-value = 0.3356

> names(cq)

[1] "statistic" "parameter" "p.value"   "method"    "data.name" "observed" 
[7] "expected"  "residuals" "stdres"   

> cq$p.value

[1] 0.3355516

Note that does the same test as prop.test, for a 2x2 table.

> chisq.test(tab)

    Pearson's Chi-squared test with Yates' continuity correction

data:  tab
X-squared = 0.92735, df = 1, p-value = 0.3356

> prop.test(tab)

    2-sample test for equality of proportions with continuity
    correction

data:  tab
X-squared = 0.92735, df = 1, p-value = 0.3356
alternative hypothesis: two.sided
95 percent confidence interval:
 -0.005208049  0.001673519
sample estimates:
   prop 1    prop 2 
0.9745028 0.9762701 

fisher.test() performs contingency table test using the hypogeometric distribution (used for small sample sizes).

fisher.test(x, y = NULL, workspace = 200000, hybrid = FALSE,
            control = list(), or = 1, alternative = "two.sided",
            conf.int = TRUE, conf.level = 0.95,
            simulate.p.value = FALSE, B = 2000)

> fisher.test(tab)

    Fisher's Exact Test for Count Data

data:  tab
p-value = 0.3324
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
 0.8001727 1.0742114
sample estimates:
odds ratio 
 0.9289923 

Sometimes you want to generate data from a distribution (such as normal), or want to see where a value falls in a known distribution. R has these distibutions built in:

• Normal
• Binomial
• Beta
• Exponential
• Gamma
• Hypergeometric
• etc

Each has 4 options:

• r for random number generation [e.g. rnorm()]
• d for density [e.g. dnorm()]
• p for probability [e.g. pnorm()]
• q for quantile [e.g. qnorm()]

> rnorm(5)

[1]  0.41760893 -0.00351707  0.47694857  2.00441824  0.60630585

The sample() function is pretty useful for permutations

> sample(1:10, 5, replace=FALSE)

[1] 2 1 4 7 8

Also, if you want to only plot a subset of the data (for speed/time or overplotting)

> samp.cars <- cars[ sample(nrow(cars), 10000), ]
> plot(VehOdo ~ jitter(VehBCost,amount=0.3), data= samp.cars)