# Introduction to Statistical Analysis

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As a neophyte researcher performing a quantitative study, you are confronted with many critical decisions related to data analysis that require statistical knowledge. To perform statistical analysis of data from a quantitative study, you need to be able to (1) determine the necessary sample size to power your study adequately; (2) prepare the data for analysis; (3) describe the sample; (4) test the reliability of measures used in the study; (5) perform exploratory analyses of the data; (6) perform analyses guided by the study objectives, questions, or hypotheses; and (7) interpret the results of statistical procedures. We recommend consulting with a statistician or expert researcher early in the research process to help you develop a plan for accomplishing these seven tasks. A statistician is also invaluable in conducting data analysis for a study and interpreting the results.

Critical appraisal of the results of studies and statistical analyses both require an understanding of the statistical theory underlying the process of analysis. This chapter and the following four chapters provide you with the information needed for critical appraisal of the results sections of published studies and for performance of statistical procedures to analyze data in studies and in clinical practice. This chapter introduces the concepts of statistical theory and discusses some of the more pragmatic aspects of quantitative data analysis: the purposes of statistical analysis, the process of performing data analysis, the method for choosing appropriate statistical analysis techniques for a study, and resources for conducting statistical analysis procedures. Chapter 22 explains the use of statistics for descriptive purposes, such as describing the study sample or variables. Chapter 23 focuses on the use of statistics to examine proposed relationships among study variables, such as the relationships among the variables dyspnea, anxiety, and quality of life. Chapter 24 explores the use of statistics for prediction, such as using independent variables of age, gender, cholesterol values, and history of hypertension to predict the dependent variable of cardiac risk level. Chapter 25 guides you in using statistics to determine differences between groups, such as determining the difference in muscle strength and falls (dependent variables) between an experimental or intervention group receiving a strength training program (independent variable) and a comparison group receiving standard care.

## Concepts of Statistical Theory

One reason nurses tend to avoid statistics is that many were taught the mathematical mechanics of calculating statistical formulas and were given little or no explanation of the logic behind the analysis procedure or the meaning of the results (Grove, 2007). This mathematical process is usually performed by computer, and information about it offers little assistance to the individuals making statistical decisions or explaining results. We approach data analysis from the perspective of enhancing your understanding of the meaning underlying statistical analysis. You can use this understanding either for critical appraisal of studies or for conducting data analyses.

The ensuing discussion explains some of the concepts commonly used in statistical theory. The logic of statistical theory is embedded within the explanations of these concepts. The concepts presented in this chapter include probability theory, classical hypothesis testing, Type I and Type II errors, statistical power, statistical significance versus clinical importance, inference, samples and populations, descriptive and inferential statistical techniques, measures of central tendency, the normal curve, sampling distributions, symmetry, skewness, modality, kurtosis, variation, confidence intervals, and parametric and nonparametric types of inferential statistical analyses.

### 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.

1. State your primary null hypothesis. (Chapter 8 discusses the development of the null hypothesis.)

2. Set your study alpha (Type I error); this is usually α = 0.05.

3. Set your study beta (Type II error); this is usually β = 0.20.

4. Conduct power analyses (Aberson, 2010; Cohen, 1988).

5. Design and conduct your study.

6. Compute the appropriate statistic on your obtained data.

7. Compare your obtained statistic with its corresponding theoretical distribution in the tables provided in the Appendices at the back of this book. For example, if you analyzed your data with a *t*-test, you would compare the *t* value from your study with the critical values of *t* in the table.

8. If your obtained statistic exceeds the critical value in the distribution table, you can reject your null hypothesis. If not, you must accept your null hypothesis. These ideas are discussed in more depth in Chapters 23 through 25 when the results of statistics are presented.

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.

TABLE 21-1

Decision | |||

Reject Null | Accept Null | ||

True Population Status | Null Is True |
Type I Errorα | Correct Decision1 − α |

Null Is False |
Correct Decision1 − β | Type II Error β |

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 Power

**Power** is the probability that a statistical test will detect an effect when it actually exists. Power is the inverse of Type II error and is calculated as 1 − β. Type II error is the probability of retaining the null hypothesis when it is in fact false. When the researcher sets Type II error at 0.20 before conducting a study, this means that the power of the planned statistic has been set to 0.80. In other words, the statistic will have an 80% chance of detecting an effect if it actually exists.

Reported studies failing to reject the null hypothesis (in which power is unlikely to have been examined) often have a low power level to detect an effect if one exists. Until more recently, the researcher’s primary interest was in preventing a Type I error. Therefore, great emphasis was placed on the selection of a level of significance, but little emphasis was placed on power. This point of view is changing as we recognize the seriousness of a Type II error in nursing studies.

As stated in the steps of classical hypothesis testing previously, step 4 is “conducting a power analysis.” Power analysis involves determining the required sample size needed to conduct your study after performing steps 1, 2, and 3. Cohen (1988) identified four parameters of power: (1) significance level, (2) sample size, (3) effect size, and (4) power (standard of 0.80). If three of the four are known, the fourth can be calculated by using power analysis formulas. Significance level and sample size are straightforward. Chapter 15 provides a detailed discussion of determining sample size in quantitative studies that includes power analysis. **Effect size** is “the degree to which the phenomenon is present in the population or the degree to which the null hypothesis is false” (Cohen, 1988, pp. 9-10). For example, suppose you were measuring changes in anxiety levels, measured first when the patient is at home and then just before surgery. The effect size would be large if you expected a great change in anxiety. If you expected only a small change in the level of anxiety, the effect size would be small.

Small effect sizes require larger samples to detect these small differences (see Chapter 15 for a detailed discussion of effect size). If the power is too low, it may not be worthwhile conducting the study unless a large sample can be obtained because statistical tests are unlikely to detect differences or relationships that exist. Deciding to conduct a study in these circumstances is costly in time and money, frequently does not add to the body of nursing knowledge, and can lead to false conclusions. Power analysis can be conducted via hand calculations, computer software, or online calculators and should be performed to determine the sample size necessary for a particular study (Aberson, 2010). Power analysis can be calculated by using the free power analysis software G*Power (Faul, Erdfelder, Lang, & Buchner, 2007) or statistical software such as NCSS, SAS, and SPSS (Table 21-2). In addition, many free sample size calculators are available online that are easy to use and understand. If you have questions, you could consult a statistician.

TABLE 21-2

Software Applications for Statistical Analysis

Software Application | Website |

NCSS (Number Cruncher Statistical System) | www.ncss.com |

SPSS (Statistical Packages for the Social Sciences) | www.spss.com |

SAS (Statistical Analysis System) | www.sas.com |

S+ | spotfire.tibco.com |

Stata | www.stat.com |

JMP | www.jmp.com |

The power achieved should be reported with the results of the studies, especially studies that fail to reject the null hypothesis (have nonsignificant results). If power is high, it strengthens the meaning of the findings. If power is low, researchers need to address this issue in the discussion of limitations and implications of the study findings. Modifications in the research methodology that resulted from the use of power analysis also need to be reported.

### 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.