Probability Theory

Probability theory addresses statistical analysis as the likelihood of accurately predicting an event or the extent of an effect. Nurse researchers might be interested in the probability of a particular nursing outcome in a particular patient care situation. For example, what is the probability of patients older than 75 years of age with cardiac conditions falling when hospitalized? With probability theory, you could determine how much of the variation in your data could be explained by using a particular statistical analysis. In probability theory, the researcher interprets the meaning of statistical results in light of his or her knowledge of the field of study. A finding that would have little meaning in one field of study might be important in another (Good, 1983; Kerlinger & Lee, 2000). Probability is expressed as a lowercase p, with values expressed as percentages or as a decimal value ranging from 0 to 1. For example, if the exact probability is known to be 0.23, it would be expressed as p = 0.23. The p in statistics is defined as the probability of rejecting the null hypothesis when the null is actually true. Nurse researchers typically consider a p = 0.05 value or less to indicate a real effect.

Classical Hypothesis Testing

Classical hypothesis testing refers to the process of testing a hypothesis to infer the reality of an effect. This process starts with the statement of a null hypothesis, which assumes no effect (e.g., no difference between groups, or no relationship between variables). The researcher sets the values of two theoretical probabilities: (1) the probability of rejecting the null hypothesis when it is in fact true (alpha [α], Type I error) and (2) the probability of retaining the null hypothesis when it is in fact false (beta [β], Type II error). In nursing research, alpha is usually set at 0.05, meaning that the researcher will allow a 5% or lower chance of making a Type I error. The beta is frequently set at 0.20, meaning that the researcher will allow for a 20% or lower chance of making a Type II error.

After conducting the study, the researcher culminates the hypothesis testing process by making a rational decision either to reject or to retain the null hypothesis, based on the statistical results. The following steps outline each of the components of statistical hypothesis testing.

Cox (1958, p. 159) stated, “Significance tests, from this point of view, measure the adequacy of the data to support the qualitative conclusion that there is a true effect in the direction of the apparent difference.” Thus, the decision is a judgment and can be in error. The level of statistical significance attained indicates the degree of uncertainty in taking the position that the difference between the two groups is real. Classical hypothesis testing has been largely criticized for such errors in judgments (Cohen, 1994; Loftus 1993). Much emphasis has been placed on researchers providing indicators of effect, rather than just relying on p values, specifically, providing the magnitude of the obtained effect (e.g., a difference or relationship) as well as confidence intervals associated with the statistical findings. These additional statistics give consumers of research more information about the phenomenon being studied (Cohen 1994).

Type I and Type II Errors

We choose the probability of making a Type I error when we set alpha, and if we decrease the probability of making a Type I error, we increase the probability of making a Type II error. The relationships between Type I and Type II errors are defined in Table 21-1. Type II error occurs as a result of some degree of overlap between the values of different populations, so in some cases a value with a greater than 5% probability of being within one population may be within the dimensions of another population.

It is impossible to decrease both types of error simultaneously without a corresponding increase in sample size. The researcher needs to decide which risk poses the greatest threat within a specific study. In nursing research, many studies are conducted with small samples and instruments that lack precision and accuracy in the measurement of study variables. Many nursing situations include multiple variables that interact to lead to differences within populations. However, when one is examining only a few of the interacting variables, small differences can be overlooked and could lead to a false conclusion of no differences between the samples. In this case, the risk of a Type II error is a greater concern, and a more lenient level of significance is in order. Nurse researchers usually set the level of significance or α = 0.05 for their studies versus a more stringent α = 0.01 or 0.001. Setting α = 0.05 reduces the risk of a Type II error of indicating study results are not significant when they are.

Statistical Significance versus Clinical Importance

The findings of a study can be statistically significant but may not be clinically important. For example, one group of patients might have a body temperature 0.1° F higher than that of another group. Data analysis might indicate that the two groups are statistically significantly different. However, the findings have little or no clinical importance because of the small difference in temperatures between groups. It is often important to know the magnitude of the difference between groups in studies. However, a statistical test that indicates significant differences between groups (e.g., a t-test) provides no information on the magnitude of the difference. The extent of the level of significance (0.01 or 0.0001) tells you nothing about the magnitude of the difference between the groups or the relationship between two variables. The magnitude of group differences can best be determined through calculating effect sizes and confidence intervals (see Chapters 22 through 25).

Inference

Statisticians use the terms inference and infer in a similar way that a researcher uses the term generalize. Inference requires the use of inductive reasoning. One infers from a specific case to a general truth, from a part to the whole, from the concrete to the abstract, from the known to the unknown. When using inferential reasoning, you can never prove things; you can never be certain. However, one of the reasons for the rules that have been established with regard to statistical procedures is to increase the probability that inferences are accurate. Inferences are made cautiously and with great care. Researchers use inferences to infer from the sample in their study to the larger population.

Samples and Populations

Use of the terms statistic and parameter can be confusing because of the various populations referred to in statistical theory. A statistic, such as a mean (), is a numerical value obtained from a sample. A parameter is a true (but unknown) numerical characteristic of a population. For example, µ is the population mean or arithmetic average. The mean of the sampling distribution (mean of samples’ means) can also be shown to be equal to µ. A numerical value that is the mean () of the sample is a statistic; a numerical value that is the mean of the population (µ) is a parameter (Barnett, 1982).

Relating a statistic to a parameter requires an inference as one moves from the sample to the sampling distribution and then from the sampling distribution to the population. The population referred to is in one sense real (concrete) and in another sense abstract. These ideas are illustrated as follows:

For example, perhaps you are interested in the cholesterol levels of women in the United States. Your population is women in the United States. You cannot measure the cholesterol level of every woman in the United States; therefore, you select a sample of women from this population. Because you wish your sample to be as representative of the population as possible, you obtain your sample by using random sampling techniques (see Chapter 15). To determine whether the cholesterol levels in your sample are similar to those in the population, you must compare the sample with the population. One strategy would be to compare the mean of your sample with the mean of the entire population. However, it is highly unlikely that you know the mean of the entire population; you must make an estimate of the mean of that population. You need to know how good your sample statistics are as estimators of the parameters of the population. First, you make some assumptions. You assume that the mean scores of cholesterol levels from multiple, randomly selected samples of this population would be normally distributed. This assumption implies another assumption: that the cholesterol levels of the population will be distributed according to the theoretical normal curve—that difference scores and standard deviations can be equated to those in the normal curve. The normal curve is discussed later in this chapter.

If you assume that the population in your study is normally distributed, you can also assume that this population can be represented by a normal sampling distribution. You infer from your sample to the sampling distribution, the mathematically developed theoretical population made up of parameters such as the mean of means and the standard error. The parameters of this theoretical population are the measures of the dimensions identified in the sampling distribution. You can infer from the sampling distribution to the population. You have both a concrete population and an abstract population. The concrete population consists of all the individuals who meet your study sample criteria, whereas the abstract population consists of individuals who will meet your sample criteria in the future or the groups addressed theoretically by your framework.

Types of Statistics

There are two major classes of statistics: descriptive statistics and inferential statistics. Descriptive statistics are computed to reveal characteristics of the sample and to describe study variables. Inferential statistics are computed to draw conclusions and make inferences about the greater population, based on the sample data set. The following sections define the concepts and rationale associated with descriptive and inferential statistics.

Descriptive Statistics

A basic yet important way to begin describing a sample is to create a frequency distribution of the variable or variables being studied. A frequency distribution is a plot of one variable, whereby the x-axis consists of the possible values of that variable, and the y-axis is the tally of each value. For example, if you assessed a sample for a variable such as pain using a visual analogue scale, and your subjects reported particular values for pain, you could create a frequency distribution as illustrated in Figure 21-1.

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