Scot H Simpson, BSP, PharmD, MSc, is Professor and Associate Dean, Research and Graduate Studies, Faculty of Pharmacy and Pharmaceutical Sciences, University of Alberta, Edmonton, Alberta. He is also an Associate Editor with the CJHP.
Address correspondence to: Scot H Simpson, Faculty of Pharmacy and Pharmaceutical Sciences, 3126 Dentistry/Pharmacy, University of Alberta, Edmonton AB T6G 2N8, e-mail: ac.atreblau@tocs
Copyright 2015 Canadian Society of Hospital Pharmacists. All content in the Canadian Journal of Hospital Pharmacy is copyrighted by the Canadian Society of Hospital Pharmacy. In submitting their manuscripts, the authors transfer, assign, and otherwise convey all copyright ownership to CSHP.
– Mark Twain 1There are three kinds of lies: lies, damned lies, and statistics.
Statistics represent an essential part of a study because, regardless of the study design, investigators need to summarize the collected information for interpretation and presentation to others. It is therefore important for us to heed Mr Twain’s concern when creating the data analysis plan. In fact, even before data collection begins, we need to have a clear analysis plan that will guide us from the initial stages of summarizing and describing the data through to testing our hypotheses.
The purpose of this article is to help you create a data analysis plan for a quantitative study. For those interested in conducting qualitative research, previous articles in this Research Primer series have provided information on the design and analysis of such studies. 2 , 3 Information in the current article is divided into 3 main sections: an overview of terms and concepts used in data analysis, a review of common methods used to summarize study data, and a process to help identify relevant statistical tests. My intention here is to introduce the main elements of data analysis and provide a place for you to start when planning this part of your study. Biostatistical experts, textbooks, statistical software packages, and other resources can certainly add more breadth and depth to this topic when you need additional information and advice.
When analyzing information from a quantitative study, we are often dealing with numbers; therefore, it is important to begin with an understanding of the source of the numbers. Let us start with the term variable, which defines a specific item of information collected in a study. Examples of variables include age, sex or gender, ethnicity, exercise frequency, weight, treatment group, and blood glucose. Each variable will have a group of categories, which are referred to as values, to help describe the characteristic of an individual study participant. For example, the variable “sex” would have values of “male” and “female”.
Although variables can be defined or grouped in various ways, I will focus on 2 methods at this introductory stage. First, variables can be defined according to the level of measurement. The categories in a nominal variable are names, for example, male and female for the variable “sex”; white, Aboriginal, black, Latin American, South Asian, and East Asian for the variable “ethnicity”; and intervention and control for the variable “treatment group”. Nominal variables with only 2 categories are also referred to as dichotomous variables because the study group can be divided into 2 subgroups based on information in the variable. For example, a study sample can be split into 2 groups (patients receiving the intervention and controls) using the dichotomous variable “treatment group”. An ordinal variable implies that the categories can be placed in a meaningful order, as would be the case for exercise frequency (never, sometimes, often, or always). Nominal-level and ordinal-level variables are also referred to as categorical variables, because each category in the variable can be completely separated from the others. The categories for an interval variable can be placed in a meaningful order, with the interval between consecutive categories also having meaning. Age, weight, and blood glucose can be considered as interval variables, but also as ratio variables, because the ratio between values has meaning (e.g., a 15-year-old is half the age of a 30-year-old). Interval-level and ratio-level variables are also referred to as continuous variables because of the underlying continuity among categories.
As we progress through the levels of measurement from nominal to ratio variables, we gather more information about the study participant. The amount of information that a variable provides will become important in the analysis stage, because we lose information when variables are reduced or aggregated—a common practice that is not recommended. 4 For example, if age is reduced from a ratio-level variable (measured in years) to an ordinal variable (categories of < 65 and ≥ 65 years) we lose the ability to make comparisons across the entire age range and introduce error into the data analysis. 4
A second method of defining variables is to consider them as either dependent or independent. As the terms imply, the value of a dependent variable depends on the value of other variables, whereas the value of an independent variable does not rely on other variables. In addition, an investigator can influence the value of an independent variable, such as treatment-group assignment. Independent variables are also referred to as predictors because we can use information from these variables to predict the value of a dependent variable. Building on the group of variables listed in the first paragraph of this section, blood glucose could be considered a dependent variable, because its value may depend on values of the independent variables age, sex, ethnicity, exercise frequency, weight, and treatment group.
Statistics are mathematical formulae that are used to organize and interpret the information that is collected through variables. There are 2 general categories of statistics, descriptive and inferential. Descriptive statistics are used to describe the collected information, such as the range of values, their average, and the most common category. Knowledge gained from descriptive statistics helps investigators learn more about the study sample. Inferential statistics are used to make comparisons and draw conclusions from the study data. Knowledge gained from inferential statistics allows investigators to make inferences and generalize beyond their study sample to other groups.
Before we move on to specific descriptive and inferential statistics, there are 2 more definitions to review. Parametric statistics are generally used when values in an interval-level or ratio-level variable are normally distributed (i.e., the entire group of values has a bell-shaped curve when plotted by frequency). These statistics are used because we can define parameters of the data, such as the centre and width of the normally distributed curve. In contrast, interval-level and ratio-level variables with values that are not normally distributed, as well as nominal-level and ordinal-level variables, are generally analyzed using nonparametric statistics.
The first step in a data analysis plan is to describe the data collected in the study. This can be done using figures to give a visual presentation of the data and statistics to generate numeric descriptions of the data.
Selection of an appropriate figure to represent a particular set of data depends on the measurement level of the variable. Data for nominal-level and ordinal-level variables may be interpreted using a pie graph or bar graph. Both options allow us to examine the relative number of participants within each category (by reporting the percentages within each category), whereas a bar graph can also be used to examine absolute numbers. For example, we could create a pie graph to illustrate the proportions of men and women in a study sample and a bar graph to illustrate the number of people who report exercising at each level of frequency (never, sometimes, often, or always).
Interval-level and ratio-level variables may also be interpreted using a pie graph or bar graph; however, these types of variables often have too many categories for such graphs to provide meaningful information. Instead, these variables may be better interpreted using a histogram. Unlike a bar graph, which displays the frequency for each distinct category, a histogram displays the frequency within a range of continuous categories. Information from this type of figure allows us to determine whether the data are normally distributed. In addition to pie graphs, bar graphs, and histograms, many other types of figures are available for the visual representation of data. Interested readers can find additional types of figures in the books recommended in the “Further Readings” section.
Figures are also useful for visualizing comparisons between variables or between subgroups within a variable (for example, the distribution of blood glucose according to sex). Box plots are useful for summarizing information for a variable that does not follow a normal distribution. The lower and upper limits of the box identify the interquartile range (or 25th and 75th percentiles), while the midline indicates the median value (or 50th percentile). Scatter plots provide information on how the categories for one continuous variable relate to categories in a second variable; they are often helpful in the analysis of correlations.
In addition to using figures to present a visual description of the data, investigators can use statistics to provide a numeric description. Regardless of the measurement level, we can find the mode by identifying the most frequent category within a variable. When summarizing nominal-level and ordinal-level variables, the simplest method is to report the proportion of participants within each category.
The choice of the most appropriate descriptive statistic for interval-level and ratio-level variables will depend on how the values are distributed. If the values are normally distributed, we can summarize the information using the parametric statistics of mean and standard deviation. The mean is the arithmetic average of all values within the variable, and the standard deviation tells us how widely the values are dispersed around the mean. When values of interval-level and ratio-level variables are not normally distributed, or we are summarizing information from an ordinal-level variable, it may be more appropriate to use the nonparametric statistics of median and range. The first step in identifying these descriptive statistics is to arrange study participants according to the variable categories from lowest value to highest value. The range is used to report the lowest and highest values. The median or 50th percentile is located by dividing the number of participants into 2 groups, such that half (50%) of the participants have values above the median and the other half (50%) have values below the median. Similarly, the 25th percentile is the value with 25% of the participants having values below and 75% of the participants having values above, and the 75th percentile is the value with 75% of participants having values below and 25% of participants having values above. Together, the 25th and 75th percentiles define the interquartile range.
One caveat about the information provided in this section: selecting the most appropriate inferential statistic for a specific study should be a combination of following these suggestions, seeking advice from experts, and discussing with your co-investigators. My intention here is to give you a place to start a conversation with your colleagues about the options available as you develop your data analysis plan.
There are 3 key questions to consider when selecting an appropriate inferential statistic for a study: What is the research question? What is the study design? and What is the level of measurement? It is important for investigators to carefully consider these questions when developing the study protocol and creating the analysis plan. The figures that accompany these questions show decision trees that will help you to narrow down the list of inferential statistics that would be relevant to a particular study. Appendix 1 provides brief definitions of the inferential statistics named in these figures. Additional information, such as the formulae for various inferential statistics, can be obtained from textbooks, statistical software packages, and biostatisticians.
The first step in identifying relevant inferential statistics for a study is to consider the type of research question being asked. You can find more details about the different types of research questions in a previous article in this Research Primer series that covered questions and hypotheses. 5 A relational question seeks information about the relationship among variables; in this situation, investigators will be interested in determining whether there is an association ( Figure 1 ). A causal question seeks information about the effect of an intervention on an outcome; in this situation, the investigator will be interested in determining whether there is a difference ( Figure 2 ).