Types of data

Before moving onto any kind of statistical analysis, it is important to understand the distinction between three main categories of data: these will determine the most appropriate statistical test. They are (a) continuous, (b) categorical and (c) time-to-event data.

Continuous data can be derived by adding, subtracting, multiplying and dividing. Examples of this kind of data include age, body mass index (BMI) and General Health Questionnaire (GHQ) score. Moreover, continuous data can be extended to count data such as number of epileptic episodes.

Categorical data can be: (a) binary, i.e. data with two categories, such as dead or alive, treatment or control, disease or non-disease; (b) nominal, i.e. data with categories that cannot be ordered, such as blood type, marital status or occupation; or (c) ordinal, i.e. data with ordered categories, such as school grades, age categories (e.g. 25–35, 35–45, etc.) and ratings on a Likert scale (e.g. strongly agree, agree, disagree, etc.).

Time-to-event data measure the time it takes for an event to occur, such as time to death or time to remission.

This article focuses on the analysis of continuous data.

Summary or descriptive statistics

The first step of any statistical analysis is to describe the population of interest. This will give information about patients’ demographic and clinical characteristics, such as age, gender and educational status. The best way of doing this is to produce summary statistics.

Table 1 shows the summary statistics produced on three measures of ability collected as part of the Synthetic Aperture Personality Assessment (SAPA) project (http://sapa-project.org). These are: the self-assessment SATFootnote a Verbal (SATV) score, the self-assessment SAT Quantitative (SATQ) score and the Assessment Composite Test (ACT) score (Reference Revelle, Wilt, Rosenthal, Gruszka, Matthews and SzymuraRevelle 2010). As can be seen from Table 1, gender (a binary categorical variable) is summarised as frequencies (n) and percentage proportions (%), whereas continuous variables such as age, ACT, SATV and SATQ scores are summarised by the mean, along with the standard deviation (s.d.) and/or the median, minimum (min) and maximum (max) and interquartile range (IQR).

TABLE 1

Summary statistics from the Synthetic Aperture Personality Assessment (SAPA) project

Means, medians, standard deviations and interquartile ranges

The mean value corresponds to the average value of a continuous variable, whereas the median corresponds to the exact middle value of its distribution. That is, if we rank all patients from youngest to oldest, the median value will be the one in the middle of this ranking. This is possible when we have an odd number of observations. If there is an even number of observations, the median is derived as the average of the two values in the middle of the distribution.

In some cases, the median is reported instead of the mean. The reason for this is that the latter is influenced by extreme values (outliers), which are very large or very small values. An example is shown in Fig. 1, where there are many individuals of an older age that push the value of the mean to the right of the distribution. In this case, the mean value is greater than the median and we say that the distribution is skewed to the right (it has a right tail). If the mean was smaller than the median, we would say that the distribution is skewed to the left (it would have a left tail).

FIG. 1

Histogram showing the age distribution of participants summarised in Table 1.

Similar to the mean, the standard deviation is also influenced by extreme values. The standard deviation measures the variability of a trait. So, in cases where our distribution is skewed, it is better to report the IQR instead of the standard deviation.

The IQR is the difference between the 1st and 3rd quartiles of the distribution. The 1st quartile corresponds to the value for which 25% of the observations are lower, and the 3rd quartile is the value for which 75% of the observations are lower. These values will be the same regardless of how low or high the minimum and maximum values are. Hence, the IQR is not affected by outliers.

Normality of data

If a variable is distributed symmetrically around its mean (i.e. the plotted distribution is bell-shaped), then this variable is said to follow a normal distribution. Hence, using a histogram is one way of assessing the normality of a variable.

Another way of assessing normality is by using a box plot (Fig. 2). The box plot consists of the median value (the horizontal line inside the box); the 1st and 3rd quartiles of the distribution are the lower and upper edges of the box respectively, and the minimum and maximum values of the distribution are the upper and lower ends of the plot respectively. The red dots outside the boxes are the outliers of the distribution. If the median is located in the middle of the box and the spread of the distribution is symmetrical around the median, then this is an indication of a normal distribution.

FIG. 2

Distribution of mean SAT Verbal (SATV) scores over males and females (data from Table 1).

A third way to assess normality is by a normal probability plot. If this is linear then the normality assumption is satisfied. In our case (Fig. 3) it is far from linear, which confirms our conclusion (from Fig. 1) that age was non-normally distributed.

FIG. 3

Normal probability plot for age (data from Table 1).

There are also formal tests for assessing normality, such as the Kolmogorov–Smirnov test. These tests test the null hypothesis that the distribution is normal, hence small P-values (described in the next section) suggest strong evidence against normality. Such tests, however, should not be trusted much because they are influenced by the sample size. For instance, a small sample may fail to reject a hypothesis of normality, whereas a very large sample may falsely reject a hypothesis of normality.

Box 1 gives key points on summary statistics.

BOX 1Summary or descriptive statistics

• Summary statistics are used to describe the population of interest

• The mean and standard deviation (s.d.) are reported for continuous variables whose distribution is not skewed

• The median, interquartile range (IQR), minimum and maximum values are reported for continuous variables whose distribution is skewed

• Frequencies and percentage proportions are reported for categorical variables

• Histograms, box plots and normal probability plots are used to assess the normality of data

• One should not rely solely on formal tests for assessing normality, since these are influenced by the sample size

Statistical tests

Statistical tests are used to formally test a hypothesis or, in other words, our research question. A statistical test depends on (a) the type of data, (b) the number of samples tested and (c) whether these samples are independent or correlated.

Correlation

Correlation measures the strength of a linear association. As with t -tests and ANOVA, correlation is also a special case of linear regression and it makes the same assumptions: (a) normally distributed variables, (b) equal variances and (c) continuous and independent variables. A measure of this strength is either (a) Pearson's r correlation coefficient for normally distributed outcomes or (b) Spearman's rho (ρ or r _s) correlation coefficient for non-normally distributed outcomes (and small sample size).

There is a rule of thumb with regard to the strength of an association:

• if the correlation coefficient (r or rho) is greater than 0.7, there is a strong association between the two variables

• if the correlation coefficient is between 0.3 and 0.7, there is a moderate association between the two variables

• if the correlation coefficient is between 0.1 and 0.3, there is a weak association between the two variables.

Correlation coefficients close to zero (less than 0.10) suggest no association, i.e. the two variables are independent.

Moreover, the sign of the correlation coefficient provides information about the direction of the association. A plus (+) sign would suggest a positive association, i.e. both variables increase or decrease in the same direction. A minus (−) sign would suggest a negative association, i.e. the two variables increase or decrease in opposite directions.

A scatter plot is used to explore the association between two variables, as shown in the following example. Figure 4 shows the association of heights between 928 adult children and their parents (205 mother and father pairs) almost a century and a half ago (Reference GaltonGalton 1886). It reveals a positive association between a child's height and the combined mean height of their mother and father, which Galton calls the mid-parent height (i.e. they both increase). However, a lot of observations are scattered around the line, suggesting a weak association.

FIG. 4

Scatter plot showing the association of heights between 205 couples (mother and father) and their 928 adult children. The mid-parent height is the average of the heights of the mother and father (Reference GaltonGalton 1886).

To calculate the strength of this association, we would use Pearson's correlation coefficient because (a) heights for both groups are normally distributed (and the sample is >100), (b) the variances are equal (Pearson's correlation coefficient is robust against this assumption when the sample is large enough) and (c) both variables are independent.

Using Pearson's formula on Galton's data, I found a correlation coefficient of 0.43, suggesting a moderate association between the heights of the two groups. This association, however, was statistically significant, with a corresponding P-value of less than 0.05.

Another use of Pearson's correlation coefficient is to explain variability in one variable by that in another. For instance, to calculate the variability in the children's heights explained by the mid-parent height, we square the Pearson's correlation coefficient to get the coefficient of determination, R-squared (R ²). In our example, squaring the 0.43 gives us 0.18 or 18%, suggesting that 18% of the variability in the children's heights is explained by the mid-parent heights. This leaves 82% of the remaining variability explained by factors other than the mid-parent heights.

Box 4 gives key points on correlation.

Regression

Another way to assess the association between two or more variables is with regression. There are two types of regression: (a) simple linear regression and (b) multiple linear regression. The former refers to the association between two variables, whereas the latter refers to the association between more than two variables.

Simple linear regression

Simple linear regression is very similar to the correlation described above. In the case of correlation, however, we do not distinguish which of the two variables is the dependent and which the independent variable. Since regression is eventually used for prediction, we define one variable to be the dependent (or outcome) variable and the other one to be the independent (or predictor) variable.

Simple linear regression makes the following assumptions:

• the relationship between the outcome and the predictor is linear

• the outcome is normally distributed

• the outcome has the same variance across all values of the predictor

• outcome and predictor variables are drawn from independent samples.

In the Galton example, to predict a child's height given the mid-parent height, we could do a simple linear regression, with the child's height as the dependent (or outcome) variable and the mid-parent height as the independent (or predictor) variable. The linear regression model is robust against the normality and equal variance assumption for large enough samples, and also both variables are independent, so we would fit a linear regression line (as shown in Fig. 4) to explain the association between the two variables. To express this relationship mathematically, the derived equation from fitting the model would be:

$\text{child's height}=24+0.64\times \text{mid-parent height}$

where 24 is the intercept, i.e. the average child's height if the mid-parent height is zero (as this is not possible, this value is just the average child's height in our sample); 0.64 is the regression coefficient beta (b), i.e. the slope of the regression line. The slope represents the steepness of the line: in this case by how much the child's height would change if the mid-parent height were to increase by 1 inch. For Galton's data, the beta coefficient is statistically significant (P <0.05), suggesting that there was indeed a linear relationship between the two variables.

To test the assumptions of the linear regression, we would do a residual analysis that involves plotting:

• a histogram of the residuals, to check the normality assumption (Fig. 5)

• a scatter plot of the residuals against the predictor, to check the homogeneity of variance and the independence assumption (Fig. 6).

FIG. 5

Histogram of the residuals from Galton's data.

FIG. 6

Residuals v. the predictor (mid-parent height) from Galton's data. The mid-parent height is the average of the heights of the mother and father.

Figure 5 shows a bell-shaped histogram of the residuals, suggesting that the normality assumption of the linear regression model is justified. Figure 6 shows that the residuals are randomly scattered around zero, suggesting that the variance is constant across all values of the predictor. This satisfies the homogeneity of variance and independence assumptions.

Multiple linear regression

The analysis using a simple linear regression model is known as univariate analysis, whereas the respective analysis for multiple linear regression is known as multivariate analysis. Multiple linear regression allows us to add more than one predictor into the regression equation. Hence, a multivariate regression analysis involves:

• controlling for confounder variables

• improving the precision of the estimates

• checking for interaction between the predictors (by adding an interaction term in the regression equation; for example, the addition of an age × gender interaction term in the regression model will show whether the SATQ score changes for males and females across different age groups).

Say, for instance, that we want to see whether the SATV (verbal) score is a significant predictor of the SATQ (quantitative) score in the SAPA project. Table 2 shows the results of the univariate analysis between these two variables.

TABLE 2

Univariate association between SATQ score and SATV score

The univariate analysis suggests that the SATQ score changes by 0.66 units per unit increase in the SATV score, and this association is statistically significant. We will now add age into the model to see whether age confounds this association. As can be seen from Table 3, taking age into account attenuates the effect of the SATQ score by only 1.5%, from 0.66 to 0.65. As a rule of thumb, a variable is a confounder if it changes the outcome variable by more than 10%, regardless of its significance. This is not the case in our example, so age is not a confounder. Also, the R-squared from the univariate regression is 0.415 or 41.5%, whereas the adjusted R-squared from the multivariate regression is 0.413 or 41.3%, suggesting that adding age into the model does not explain any more of the variability in the outcome than is explained by the SATV alone.

TABLE 3

Multivariate association of age and SATV score with SATQ score

Similarly, a residual analysis shows that (a) the residuals of the multivariate regression are normally distributed (Fig. 7) and (b) the homogeneity of variance and independence of variables hold for age but not for SATV score, as there seems to be a reduction in variance with increased SATV score (Fig. 8).

FIG. 7

Histogram of the residuals (from the data in Table 3).

FIG. 8

Residuals v. linear predictors (from the data in Table 3); SATV, self-assessment SAT Verbal test.

Box 5 summarises key points on regression.

BOX 5Regression

• Simple linear regression is used to assess the association between two variables

• Multiple linear regression is used to assess the association between more than two variables

• Univariate analysis involves the testing of a hypothesis using a simple linear regression

• Multivariate analysis involves the use of a multiple linear regression in order to control for confounders, improve the precision of the estimates and check for interactions

• The regression coefficient beta is the slope (or steepness) of the regression line; it shows the change in the outcome per unit increase in the predictor

• A 95% confidence interval is a range of all plausible values within which the true population value would lie 95% of the time

• A residual analysis tests the assumptions of the linear regression. It involves the use of a histogram of the residuals for testing the assumption of normality and a scatter plot of the residuals against each predictor for testing the equal variance and independent variables assumptions

• The P-value tests the hypothesis of linear association between the outcome and the predictor. A P-value less than 0.05 would suggest a statistically significant linear association between the predictor and the outcome

Principal components analysis

The purpose of principal components analysis (PCA) is to make a data-set with a large number of variables smaller without losing much information. The derived new variables are called principal components; they are linear combinations of the original variables and explain most of the variation in the data. PCA is usually done before the regression or for exploratory purposes to aid in understanding the data or in identifying patterns in the data.

As an example, we can consider a data-set with 50 observations and four variables. These four variables represent statistics on arrests per 100 000 residents for (a) assault, (b) murder and (c) rape in 50 US states in 1973 and (d) the percentage of the population living in urban areas (Reference McNeilMcNeil 1977). If we wanted to summarise these data using fewer than four variables, we would do a PCA to find the number of principal components (i.e. linear combinations of the original four variables) explaining most of the variability in the data.

Figure 9 shows a scree plot (on the left) that I created to find the number of principal components needed to explain the most variation in the data. Plot (b) shows the cumulative proportion of variance explained by the four principal components in the data. As can be seen from Fig. 9, the first component alone explains 62% of the variation in the data; the second component explains another 25%, the third component about 9% and the fourth component about 4%. An ‘elbow’ shape on a scree plot indicates the number of principal components needed to summarise the data. In this example, the crook of the elbow is on the second component, suggesting that we discard all components after the point at which the crook starts.

FIG. 9

Scree plots showing (a) the proportion of variance explained by each of the four principal components and (b) cumulative proportion of variance explained. Data from Reference McNeilMcNeil (1977).

Hence, the first two components together explain 87% of the variation in the data, suggesting that we reduced the dimensionality of the data from four to two independent variables. Table 4 shows the resulting two new variables (principal components PC1 and PC2). Each new variable is the weighted sum of the original four variables.

TABLE 4

Results of the principal components analysis showing the weights of each variable

In Table 4, the weights, which can range from −1 to +1, indicate the relative contribution of each variable to each component. For example, murder, assault and rape have weights of 0.54, 0.58 and 0.54 respectively for the first component, whereas proportion of the population living in urban areas (urban proportion) has a weight of 0.28 for this component. These data suggest that PC1 is a measure of violence.

The second component (PC2) has high weights for urban proportion (0.87) and murder (−0.42) and lower weights for assault (−0.19) and rape (0.17). Thus, these data suggest that PC2 is a measure of urban criminality.

The advantage of PCA is that we can do a multiple regression with only two variables (PC1 and PC2) instead of four. This makes it easier to interpret the results, reduces the risk of chance findings and makes the results more robust, since a composite variable may capture better a particular trait (e.g. violence) than the original variables.

Box 6 summarises key points on principal components and PCA.

Statistical software

I produced all the figures and analyses in this article using R version 3.0.2 statistical software. R is an open source (free) software that is widely used by universities worldwide. R is accompanied by a popular and powerful graphical user interface (GUI) called Deducer, which allows users to perform statistical analyses and graphing functions without any coding. Users should first download and install R (from www.r-project.org) and then download and install Deducer (from www.deducer.org).

MCQs

Select the single best option for each question stem

1. 1 Which of the following statistics should be used to summarise normally distributed variables?

   1. a Mean and standard deviation

   2. b Median and interquartile range

   3. c Frequencies and percentages

   4. d Both a and b

   5. e Both a and c.

2. 2 A large cohort study is carried out to assess the efficacy of an experimental drug in reducing depression score when compared with placebo. What test should be carried out to assess the efficacy of the drug before and after the intervention?

   1. a t-test

   2. b Wilcoxon rank-sum test

   3. c Paired t-test

   4. d Repeated-measures ANOVA

   5. e Wilcoxon signed-rank test.

3. 3 A large observational study is carried out to assess the risk factors for schizophrenia. Which type of analysis should be used to identify statistically significant risk factors?

   1. a Univariate analysis

   2. b Correlation

   3. c Multivariate analysis

   4. d Both a and c

   5. e Both b and c.

4. 4 A large randomised clinical trial is carried out to assess the efficacy of an experimental drug in reducing depression score when compared with placebo. The trial consists of a baseline assessment and 6-month and 12-month post-baseline assessments. What type of test should be used to compare the change from baseline to 6 and 12 months post-baseline between treatment and placebo?

   1. a T-test

   2. b Wilcoxon rank-sum test

   3. c Repeated-measures ANOVA

   4. d Multiple linear regression

   5. e Both a and c

5. 5 We have a large data-set of responses to a health assessment questionnaire consisting of 20 questions. We would like to devise a shorter version that would provide the same information as the original. Which of the following analyses would it be best to use to on the data to achieve this?

   1. a Principal components analysis

   2. b Multiple regression

   3. c Simple regression

   4. d Correlation

   5. e All of the above.

MCQ answers

1

a

2

c

3

d

4

c

5

a