Assumptions for parametric tests

Normal distribution of data

The p value for parametric tests depends upon a normal sampling distribution. If the sample size is large enough and the actual sample data point value are approximately normally distributed, then the central limit theorem ensures a normally distributed sampling distribution.
In regression analysis and in general linear models, it is the errors that need to be normally distributed.

Homogeneity of variance

This refers to the need for a similarity in the variance throughout the data. This means that the variable in the populations from which the samples were taken have a similar variance in these populations.
In the case of regression, the variance of one variable should be the same as all the other variables.

Interval data

It is obvious that the data point values should be for a numerical variable and measured at this level.

Independence

Data point values for variables for different groups should be independent of each other. In regression analysis, the errors should likewise be independent.
In the rest of this post, the first two assumptions are investigated.

Visual tests

In the code below, 100 data point values for a variable named hb is created. The values are taken from a normal distribution with a mean of 15 and a standard deviation of 3. A histogram with default bin size is created to visualize the frequency distribution of the data. A kernel density estimate is provided using the line() command.

hb <- rnorm(100, mean = 15, sd = 3)
hist(hb, prob = TRUE, main = "Histogram of hemoglobin values", las = 1, xlab = "Hemoglobin")
lines(density(hb))

From the plot above it seems obvious that the data are normally distributed. The QQ plot below plots the sample quantile of each data point value against its theoretical quantile. A line is added for clarity. The closer the data point values follow the line, the more likely that our assumption has been met.

qqnorm(hb, main = "QQ plot of hemoglobin values")
qqline(hb)

The next computer variable is named crp and takes on a gamma distribution. Once again, 100 data point values are created. Following this is the accompanying histogram and QQ plot.

crp = rgamma(100, 2, 2)

hist(crp, prob = TRUE, main = "Histogram of c-reactive protein values", las = 1, xlab = "CRP")
lines(density(crp))

qqnorm(crp, main = "QQ plot of CRP values")
qqline(crp)

As expected, the visual indication is that the assumption of normality is not met.

Numerical tests

Simply describing the data point values of a variable can give a good understanding of the underlying ditribution. The summary() command returns the basic descriptive statistics, including the minimum, maximum, and the quartile values.

summary(hb)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   7.922  13.051  14.555  14.777  16.755  20.038

If the mean of the data point values is approximately the same as the median value, it can be an indication of the normality of the data.

mean(hb)

## [1] 14.77742

The desc() command in the pastecs package, gives more information. It is used below in conjuction with the round() command, with its argument digits = set to 3.

library(pastecs)

round(stat.desc(hb, basic = FALSE, norm = TRUE), digits = 3)

##       median         mean      SE.mean CI.mean.0.95          var 
##       14.555       14.777        0.276        0.548        7.622 
##      std.dev     coef.var     skewness     skew.2SE     kurtosis 
##        2.761        0.187       -0.235       -0.487       -0.417 
##     kurt.2SE   normtest.W   normtest.p 
##       -0.436        0.983        0.226

The absolute values of skewness and kurtosis are interpreted as usual. The skew.2SE and kurt.2SE need some care. They express the relevant absolute value divided by twice the standard error. For small sample sizes, values of less than -1.0 or more than +1.0 indicate a p value of less than 0.05. For values less than -1.29 or more than +1.29, a p value of about 0.01 is assumed, and so on. For very large sample sizes, a small change in standard error means a large effect on significance and in such cases these values should be greatly revised.
The normtest.W and normtest.p gives the test statistic and p value of the Shapiro-Wilk test.

The Shapiro-Wilk test can be used on its own. A p value of less than 0.05 indicates a high likelihood that the assumption for normality is NOT met. Below the hb and crp variables are passed as argument to the shapito.test() command, resulting in the same test statistic and p value as above.

shapiro.test(hb)

## 
##  Shapiro-Wilk normality test
## 
## data:  hb
## W = 0.983, p-value = 0.2261

shapiro.test(crp)

## 
##  Shapiro-Wilk normality test
## 
## data:  crp
## W = 0.89939, p-value = 1.324e-06