1 Making Medicine Count



That which is not measureable is not science. Everything else is stamp collecting. Ernest Rutheford, Chemistry Nobel 1908

While we may be biased, we want to emphasize that statistics is all about numbers, about quantifying exposures and outcomes, and applying detailed probability models to make quantitive inferences. But despite their central role in medical research, numbers are not the end-all or be-all. Typically, the most important questions arise at the bedside, when you’re prompted to ask yourself why or how? In addition, formal qualitative methods, such as opinion surveys and focus groups, are powerful investigative tools in their own right or for crystallizing hypotheses and identifying unanswered questions that become starting points for more quantitative studies.

Everything that can be counted does not necessarily count; everything that counts cannot necessarily be counted. Albert Einstein, Physics Nobel 1921

1.1 What do we mean by ‘Statistics’?

  • Statistics is the study of how to collect, organize, analyze, and interpret numerical information from data

  • Descriptive statistics involves methods of organizing, picturing and summarizing information from data.

  • Inferential statistics involves methods of using information from a sample to draw conclusions about the population.

Typically, we are interested in a populations e.g. all school-age children in Manitoba. Populations can be characterized by parameters, which are numeric summaries of measurements made on the entire population e.g. mean blood pressure (\(\mu\)) or standard deviation (\(\sigma\)), usually unknown. Population parameters are typically referred to using Greek symbols.

Since we rarely have data on an entire population, population parameters are usually estimated from sample statistics e.g. sample means, proportions, and standard deviations.

Sample statistics are point estimates and, as estimates, inevitably include some degree of uncertainty due to both measurement and sampling errors (sample-to-sample variability). As estimates, we measure their precision (uncertainty) in terms of standard errors or confidence intervals.

A POINT ESTIMATES WITHOUT A MEASURE OF PRECISION IS INCOMPLETE!!!

1.2 Random Samples

  • Simple Random Sample (SRS): every member of the population has equal probability of being included. Population parameters ≈ sample statistics

  • Stratified Sampling: take a SRS from each stratum (e.g race, sex)

  • Systematic Sampling: choose the staring point at random if population elements are arranged sequentially e.g. every \(5^{th}\) patient in the ER.

  • Cluster Sampling: randomly sample geographic clusters (e.g. schools, clinics)

  • Convenience Sampling: using data that are convenient and readily obtained

Inferences are no stronger than the data they’re built on. Ideally, we select our random sample by drawing numbers from a clinic list or telephone directory (aka the sampling frame).

Often, investigators simply assume their sample is a simple random sample or SRS, which is the most common type of sample. In an SRS, every member of the population has equal probability of being included i.e. the sample is deliberately chosen to be representative of the population, and the population estimates for means or proportions are simply their sample values. For example, the mean BP of our sample of high school students is our best estimate of the population if they were chosen as a SRS.

Stratified sampling is a another common sampling method. Here, you stratify or divide the population by race or sex and take a simple random sample from each stratum. For our blood pressure examples, we might take a sample of 100 white children and 100 aboriginal children, so as to generate race-specific normative data. And we could still estimate population parameters for the entire population by taking weighted averages, for example weighting the strata means in terms of the racial proportions in the overall population. Obviously, the mathematics becomes slightly more complicated. For those of you working with Statistics Canada survey data, you will have to deal with ‘survey weights’, that allow you to calculate weighted averages to estimate population parameters.

You need to be very careful with convenience samples, which you will often find on websites that ask visitors to fill out a poll or pollsters who select the first 100 people that will talk to them. In general, you know nothing about the population represented by the convenience sample, and you have no way of knowing how to convert sample statistics to population estimates, since you have no sampling frame and no survey weights. As a survey design, this is rather unscientific and the results do not necessarily generalize to a meaningful population

1.3 Key Elements of a Research Study

A meaningful research study usually requires:

  • A clearly defined outcome i.e. response, dependent, y variable

  • A clearly defined exposure i.e. treatment, independent, predictor, covariates, x variables

  • A clear question/ research hypothesis involving the relationship between exposure and outcome

1.4 Types of Variables

  • Continuous numeric variables (quantitative, interval, ratio). In theory, they can take on any numeric value from -\(\infty\) to \(\infty\) (infinity) e.g. BP, weight

  • Discrete numeric variables: Can only take on integer values e.g. counts 0, 1, 2, …

  • Nominal categorical variables: Arbitrary categories or ‘levels’ without intrinsic ordering e.g.

    • hair color, treatment indicator, diagnosis

    • often binary (dichotomous): 0 – 1 indicator or dummy variables

  • Ordinal categorical variables. Here, the ‘levels’ are defined on an ordered, semi-quantitative scale:

    • low – medium – high blood pressure

    • urine dipsticks (Neg, 1+, 2+, 3+)

    • Likert scores (e.g. histologic inflammation 0-3)

While these distinctions may seem pedantic, we will learn that statistical methods follow almost automatically from the types of variables being compared. In the next chapter, we will see how the type of variable dictates the appropriate graphical exploration. When we study regression or generalized linear models, we will see that each type of outcome variable has a distinct procedure associated with it e.g. 

  • Linear regression if the outcome is continuous numeric

  • Logistic regression if the outcome is dichotomous

  • Multinomial regression if the outcome is nominal categorical

  • Proportional odds logistic regression if the outcome is ordinal categorical

  • Poisson or negative binomial regression if the outcome is a count or frequency

  • Cox proportional hazards model (or accelerated failure time models) if the outcome is a censored time-to-event e.g. survival time.

etc