chapter 021

1Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.

Chapter 21

Introduction to Statistical Analysis


Why is Knowledge of Statistics Required to Understand Nursing Research?

Used by both qualitative and quantitative researchers to describe the sample

Used by quantitative researchers to analyze the results


Critical Appraisal of the Results Section of a Quantitative Study

Identify statistical procedures used Judge whether these statistical procedures

were appropriate for the hypotheses, questions, or objectives of study and for the data available for analysis


Critical Appraisal of the Results Section of a Quantitative Study (Cont’d)

Comprehend discussion of data analysis results

Judge whether author’s interpretation of the results is appropriate

Evaluate the clinical importance of the findings


The Researcher Uses Statistics to

Determine the necessary sample size to adequately power the study

Prepare the data for analysis Describe the sample


The Researcher Uses Statistics to (Cont’d)

Test the reliability of measures used in the study

Perform exploratory analyses of the data Perform analyses guided by the study

objectives, questions, or hypotheses Interpret the results of statistical procedures


So …

Consulting with a statistician or expert researcher early in the research process is a great idea. He or she will help you do the following: Develop a plan

Conduct data analysis Interpret the results


Concepts of Statistical Theory

Probability theory Classical hypothesis testing Type I and type II errors Statistical power Statistical significance versus clinical

importance Inference, samples, and populations


Concepts of Statistical Theory (Cont’d)

Descriptive and inferential statistical techniques

Measures of central tendency The normal curve Sampling distributions Symmetry Skewness


Concepts of Statistical Theory (Cont’d)

Modality Kurtosis Variation Confidence intervals Parametric and nonparametric types of

inferential statistical analyses


Probability Theory

Likelihood of accurately predicting an event or the extent of an effect

Expressed as a lowercase p Values expressed as percentages or as a

decimal value ranging from 0 to 1 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


Statistical Hypothesis Testing Steps

1. State your primary null hypothesis

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

3. Set your study beta (Type II error). It is usually = 0.20

4. Conduct power analyses

5. Design and conduct your study

6. Compute the appropriate statistic on your obtained data


Statistical Hypothesis Testing Steps (Cont’d)

7. Compare your obtained statistic with its corresponding theoretical distribution tables.

8. If your obtained statistic exceeds the critical value in the distribution table, you can reject your null hypothesis. If not, then you must accept your null hypothesis.


Type I And Type II Errors

Setting α is the same as choosing the probability of making a Type I error

Setting β is the same as choosing the probability of making a Type II error

If we decrease the probability of making a Type I error, we increase the probability of making a Type II error


Type I And Type II Errors (Cont’d)

Example: Does the experimental treatment produce different results than the control treatment does?

Truth

Null hypothesis rejected

(There IS a difference....)

Null hypothesis accepted

(There is no difference...)

Null hypothesis correct

(There really isn’t a difference)

TYPE I ERROR

Results are statistically significant but there is no

difference.

CORRECT CONCLUSION

Results are not statistically significance and there is no

difference.

Null hypothesis incorrect

(There really IS a difference.)

CORRECT CONCLUSION

Statistically significant results and there really is a

difference!

TYPE II ERROR

Results are not statistically significant but there really is a

difference.


Statistical Power

Power is the probability that a statistical test will detect an effect when it actually exists. So, power is the extent to which a researcher expects Type II error [β] Not to Occur

Calculated as 1 – β If Type II error is set at 0.20, power of test to

detect a difference is set at 0.80 So the statistic will detect a difference if it

actually exists


Tool for Power Calculation

Ruth Lenth’s page:

www.divms.uiowa.edu/~rlenth/Power/


General Rules

If an intervention is Benign, the researcher sets the alpha high (alpha [p-level] 0.05 to 0.10)

If an intervention is Dangerous, the researcher sets the alpha low (0.01, or 0.005, or even 0.0001)


Power Analysis

A power analysis determines the number of subjects the researcher will need in the study, given a certain statistical test, with a given level of significance.


Power Analysis (Cont’d)

Power depends on Sample size (larger sample, more power) Effect size (larger effect size, more power) Level of significance (p-value) (larger p-value,

more power) Statistical test used

If three of these four are known, the fourth can be calculated by using power analysis formulas.


Statistical Significance Versus Clinical Importance

The findings of a study can be statistically significant but may not be clinically important (especially with a huge sample).

Examples: weight loss in morbidly obese women, change of labor pain intensity, change in Iq scores with a pre-test workshop ...


Inference

If the results are statistically significant, the researcher makes an inference

Example: people who come from a two-language home learn a third language more easily in adulthood, so an inference might be that teaching children a second language in elementary school sets them up for learning other languages more easily.


Samples and Populations

Does the statistic of the sample equal the statistic of the population?

When may a researcher Infer that this so? When may a researcher Not Infer that this

so? If the population in the study is normally

distributed, a smaller sample may be used.


Samples and Populations (Cont’d)


Types of Statistics

Descriptive statistics describe the sample. They also may be used to draw conclusions and make inferences about the population, based on the sample

Example: the sample had a birth weight of 7 lbs. 4 oz


Types of Statistics (Cont’d)

Inferential statistics are computed to draw conclusions and make inferences about the greater population, based on the sample

Example: the sample responded positively to the intervention


Measures of Central Tendency

Descriptive statistics Identification of the center or predominant

value of a data set Mean: arithmetic average Median: exact middle value (or the average of

the middle two values if “n” is an even number)

Mode: most commonly occurring value(s)


The Normal Curve

Theoretical frequency distribution of all possible scores

Symmetrical, unimodal, and has continuous values

Mean = median = mode


Sampling Distributions

As the sample size becomes larger, the shape of the distribution will more accurately reflect the shape of the population from which the sample was taken.

As the sample size becomes larger, the distribution becomes more like the normal curve.


Central Limit Theorem

Even when statistics, such as means, come from a population with a skewed (asymmetrical) distribution, the sampling distribution developed from multiple means obtained from that skewed population will tend to fit the pattern of the normal curve.


Symmetry

Left side is a mirror image of the right side


Skewness

Any curve that is not symmetrical is referred to as skewed or asymmetrical.

Positively skewed: largest portion of data is below the mean.

Negatively skewed: largest portion of data is above the mean.

Mean, median, and mode are not equal.


Positively Skewed Distribution and a Negatively Skewed Distribution


Modality

The hump in the curve Unimodal: one mode, and frequencies

progressively decline as they move away from the mode (one hump)

Bimodal: two humps Trimodal: three humps


Bimodal Distribution


Kurtosis

Degree of peakedness of the curve Related to the spread or variance of scores Leptokurtic: extremely peaked curve Mesokurtic: intermediate degree of kurtosis Platykurtic: relatively flat curve


Kurtosis (Cont’d)


Tests of Normality

Shapiro-Wilk’s W test: assesses whether a variable’s distribution is skewed and/or kurtotic

Kolmogorov-Smirnov D test: alternative test of normality for large samples (n> 2000)


Variation

Range Variance Standard deviation [the square root of the

variance]


Confidence Intervals

The probability that a measured value will fall within a certain range

Example: the probability that heights of 30-year-old women in Sacramento will fall between 60” and 68”


Inferential Statistics

Computed to draw conclusions and make inferences about the greater population, based on the sample dataset

Two main types: parametric and non-parametric


Parametric Statistics

Most commonly used type of statistical analysis

Findings are inferred to the parameters of a normally distributed population


Parametric Statistics (Cont’d)

Require meeting the following three assumptions: Sample drawn from a population for which the

variance can be calculated Level of measurement at least interval level data

or ordinal data with an approximately normal distribution

Data treatable as random sample


Nonparametric Statistics

Distribution-free techniques Used in studies that do not meet the first two

assumptions of parametric statistics Less able to detect differences and have a

greater risk of a Type II error if the data do meet the assumptions of parametric procedures

Performed on ranks of the original data


Uses for Statistics

Summarize Explore meaning of deviations in data Compare or contrast descriptively Test proposed relationships in a theoretical

model


Uses for Statistics (Cont’d)

Infer that findings from sample are the same for the population

Examine correlation or causality Predict Infer from the sample to a theoretical model

(generalize)


Quantitative Data Analysis Stages

Preparation of data for analysis Description of sample Testing reliability of measurement Exploratory analysis of data Confirmatory analysis guided by hypotheses,

questions, or objectives Post hoc analysis


Preparing the Data for Analysis

Computers almost universally used for data analysis

Have a systematic plan for data entry Examine the data for errors Identify all missing data points Transform skewed or non-normally distributed

data to values distributed closer to normal curve


Preparing the Data for Analysis (Cont’d)

Calculated variables Backup information regularly Secure data files as designated by

institutional review board (IRB) policies Make PDF files of each output file and store

in same folder as your datasets and reports Systematically name all files


Description of the Sample

Demographic variables (age, gender, economic status, etc.) are analyzed with appropriate analysis techniques and used to develop characteristics of sample


Exploratory Analysis of the Data

Examine all the data descriptively Conduct measures of central tendency and

dispersion Examine outliers Use tables and graphs to help visualize data


Confirmatory Analysis

Performed to confirm expectations regarding data that are expressed as hypotheses, questions, or objectives

Findings are inferred from the sample to the population

Written analysis plan must describe clearly the confirmatory analyses that will be performed to examine each hypothesis, question, or objective


Confirmatory Analysis (Cont’d)

1. Identify level of measurement of data available for analysis

2. Select statistical procedure(s) appropriate for level of measurement

3. Select level of significance that will be used (usually α= 0.05)

4. Choose one-tailed versus two-tailed test, whichever is appropriate



1. Determine sample size

2. Evaluate representativeness of the sample

3. Calculate risk of a Type II error

4. Develop dummy tables and graphics to illustrate methods to be used to display the results



1. Determine degrees of freedom for the analyses

2. Perform the analyses

3. Compare values obtained with table value, using the level of significance, tailedness of the test, and df

4. Re-examine analysis

5. Interpret results in terms of hypothesis, question, and framework


Post Hoc Analysis

Commonly performed in studies with more than two groups when the analysis indicates that groups are significantly different from one another but does not indicate which of the groups is different


Choosing Appropriate Statistical Procedures for a Study

Multiple factors involved in determining suitability of a statistical procedure for a particular study The purpose of the study Hypotheses, questions, or objectives Design Level of measurement Previous experience in statistical analysis


Choosing Appropriate Statistical Procedures for a Study (Cont’d)

Statistical knowledge level Availability of statistical consultation Financial resources Access to computers

Most important factor to examine when choosing a statistical procedure is the study hypothesis

Decision tree

chapter 021

Documents

imprint of elsevier

statistical analysis

quantitative study contd

results section

objectives of study

study objectives

quantitative researchers

probability theory likelihood