Applied Univariate, Bivariate, and Multivariate Statistics Using Python. Daniel J. Denis
proportion of the general population has the virus? Ideally, researchers wanted to know how many people world-wide had contracted the virus. This constituted a case of parameter estimation, where the parameter of interest was the proportion of cases world-wide having the virus. Since this number was unknown, it was typically estimated based on sample data by computing a statistic (i.e. in this case, a proportion) and using that number to infer the true population proportion. It is important to understand that the statistic in this case was a proportion, but it could have also been a different function of the data. For example, a percentage increase or decrease in COVID-19 cases was also a parameter of interest to be estimated via sample data across a particular period of time. In all such cases, we wish to estimate a parameter based on a statistic.
What proportion of those who contracted the virus will die of it? That is, what is the estimated total death count from the pandemic, from beginning to end? Statistics such as these involved projections of death counts over a specific period of time and relied on already established model curves from similar pandemics. Scientists who study infectious diseases have historically documented the likely (i.e. read: “probabilistic”) trajectories of death rates over a period of time, which incorporates estimates of how quickly and easily the virus spreads from one individual to the next. These estimates were all statistical in nature. Estimates often included confidence limits and bands around projected trajectories as a means of estimating the degree of uncertainty in the prediction. Hence, projected estimates were in the opinion of many media types “wrong,” but this was usually due to not understanding or appreciating the limits of uncertainty provided in the original estimates. Of course, uncertainty limits were sometimes quite wide, because predicting death rates was very difficult to begin with. When one models relatively wide margins of error, one is protected, in a sense, from getting the projection truly wrong. But of course, one needs to understand what these limits represent, otherwise they can be easily misunderstood. Were the point estimates wrong? Of course they were! We knew far before the data came in that the point projections would be off. Virtually all point predictions will always be wrong. The issue is whether the data fell in line with the prediction bands that were modeled (e.g. see Figure 1.1). If a modeler sets them too wide, then the model is essentially quite useless. For instance, had we said the projected number of deaths would be between 1,000 and 5,000,000 in the USA, that does not really tell us much more than we could have guessed by our own estimates not using data at all! Be wary of “sophisticated models” that tell you about the same thing (or even less!) than you could have guessed on your own (e.g. a weather model that predicts cold temperatures in Montana in December, how insightful!).
Measurement issues were also at the heart of the pandemic (though rarely addressed by the media). What exactly constituted a COVID-19 case? Differentiating between individuals who died “of” COVID-19 vs. died “with” COVID-19 was paramount, yet was often ignored in early reports. However, the question was central to everything! “Another individual died of COVID-19” does not mean anything if we do not know the mechanism or etiology of the death. Quite possibly, COVID-19 was a correlate to death in many cases, not a cause. That is, within a typical COVID-19 death could lie a virtual infinite number of possibilities that “contributed” in a sense, to the death. Perhaps one person died primarily from the virus, whereas another person died because they already suffered from severe heart disease, and the addition of the virus simply complicated the overall health issue and overwhelmed them, which essentially caused the death.
Figure 1.1 Sample death predictions in the United States during the COVID-19 pandemic in 2020. The connected dots toward the right of the plot (beyond the break in the line) represent a point prediction for the given period (the dots toward the left are actual deaths based on prior time periods), while the shaded area represents a band of uncertainty. From the current date in the period of October 2020 forward (the time in which the image was published), the shaded area increases in order to reflect greater uncertainty in the estimate. Source: CDC (Centers for Disease Control and Prevention); Materials Developed by CDC. Used with Permission. Available at CDC (www.cdc.gov) free of charge.
To elaborate on the above point somewhat, measurement issues abound in scientific research and are extremely important, even when what is being measured is seemingly, at least at first glance, relatively simple and direct. If there are issues with how best to measure something like “COVID death,” just imagine where they surface elsewhere. In psychological research, for instance, measurement is even more challenging, and in many cases adequate measurement is simply not possible. This is why some natural scientists do not give much psychological research its due (at least in particular subdivisions of psychology), because they are doubtful that the measurement of such characteristics as anxiety, intelligence, and many other things is even possible. Self-reports are also usually fraught with difficulty as well. Hence, assessing the degree of depression present may seem trivial to someone who believes that a self-report of such symptoms is meaningless. “But I did a complex statistical analysis using my self-report data.” It doesn’t matter if you haven’t sold to the reader what you’re analyzing was successfully measured. The most important component to a house is its foundation. Some scientists would require a more definite “marker” such as a biological gene or other more physical characteristic or behavioral observation before they take your ensuing statistical analysis seriously. Statistical complexity usually does not advance a science on its own. Resolution of measurement issues is more often the paramount problem to be solved.
The key point from the above discussion is that with any research, with any scientific investigation, scientists are typically interested in estimating population parameters based on information in samples. This occurs by way of probability, and hence one can say that virtually the entire edifice of statistical and scientific inference is based on the theory of probability. Even when probability is not explicitly invoked, for instance in the case of the easy result in an experiment (e.g. 100 rats live who received COVID-19 treatment and 100 control rats die who did not receive treatment), the elements of probability are still present, as we will now discuss in surveying at a very intuitive level how classical hypothesis testing works in the sciences.
1.1 How Statistical Inference Works
Armed with some examples of the COVID-19 pandemic, we can quite easily illustrate the process of statistical inference on a very practical level. The traditional and classical workhorse of statistical inference in most sciences is that of null hypothesis significance testing (NHST), which originated with R.A. Fisher in the early 1920s. Fisher is largely regarded as the “father of modern statistics.” Most of the classical techniques used today are due to the mathematical statistics developed in the early 1900s (and late 1800s). Fisher “packaged” the technique of NHST for research workers in agriculture, biology, and other fields, as a way to grapple with uncertainty in evaluating hypotheses and data. Fisher’s contributions revolutionized how statistics are used to answer scientific questions (Denis, 2004).
Though NHST can be used in several different contexts, how it works is remarkably the same in each. A simple example will exemplify its logic. Suppose a treatment is discovered that purports to cure the COVID-19 virus and an experiment is set up to evaluate whether it does or not. Two groups of COVID-19 sufferers are recruited who agree to participate in the experiment. One group will be the control group, while the other group will receive the novel treatment. Of the subjects recruited, half will be randomly assigned to the control group, while the other half to the experimental group. This is an experimental design and constitutes the most rigorous means known to humankind for establishing the effectiveness of a treatment in science. Physicists, biologists, psychologists, and many others regularly use experimental designs in their work to evaluate potential treatment effects. You should too!
Carrying on with our example, we set up what is known as a null hypothesis,