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1Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Chapter 21
Introduction to Statistical Analysis
2Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
3Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
4Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
5Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
The Researcher Uses Statistics to
Determine the necessary sample size to adequately power the study
Prepare the data for analysis Describe the sample
6Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
7Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
8Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
9Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Concepts of Statistical Theory (Cont’d)
Descriptive and inferential statistical techniques
Measures of central tendency The normal curve Sampling distributions Symmetry Skewness
10Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Concepts of Statistical Theory (Cont’d)
Modality Kurtosis Variation Confidence intervals Parametric and nonparametric types of
inferential statistical analyses
11Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
12Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
13Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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.
14Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
15Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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.
16Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
17Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Tool for Power Calculation
Ruth Lenth’s page:
www.divms.uiowa.edu/~rlenth/Power/
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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)
19Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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.
20Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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.
21Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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 ...
22Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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.
23Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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.
24Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Samples and Populations (Cont’d)
25Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
26Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
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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)
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The Normal Curve
Theoretical frequency distribution of all possible scores
Symmetrical, unimodal, and has continuous values
Mean = median = mode
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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.
30Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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.
31Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Symmetry
Left side is a mirror image of the right side
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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.
33Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Positively Skewed Distribution and a Negatively Skewed Distribution
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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
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Bimodal Distribution
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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
37Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Kurtosis (Cont’d)
38Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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)
39Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Variation
Range Variance Standard deviation [the square root of the
variance]
40Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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”
41Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
42Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Parametric Statistics
Most commonly used type of statistical analysis
Findings are inferred to the parameters of a normally distributed population
43Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
44Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
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Uses for Statistics
Summarize Explore meaning of deviations in data Compare or contrast descriptively Test proposed relationships in a theoretical
model
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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)
47Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
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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
49Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
50Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Description of the Sample
Demographic variables (age, gender, economic status, etc.) are analyzed with appropriate analysis techniques and used to develop characteristics of sample
51Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
52Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
53Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
54Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Confirmatory Analysis (Cont’d)
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
55Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Confirmatory Analysis (Cont’d)
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
56Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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
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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
58Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
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