Statistical Tests
Karen H. Hagglund, M.S.
Karen.Hagglund@stjohn.org
Research:
How Do I Begin???
Take It “Bird by Bird”
Anne Lamott
Let’s Take it Step by Step...
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Identify topic
Literature review
Variables of interest
Research hypothesis
Design study
Power analysis
Write proposal
Design data tools
Committees

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Collect data
Set up spreadsheet
Enter data
Statistical analysis
Graphs
Slides / poster
Write paper /
manuscript
Confused by Statistics ?
Goals

To understand why a particular statistical
test was used for your research project

To interpret your results

To understand, evaluate, and present your
results
Free Statistics Software
Mystat:
http://www.systat.com/MystatProducts.aspx
List of Free Statistics Software:
http://statpages.org/javasta2.html
Before choosing a statistical
test…

Figure out the variable type
– Scales of measurement (qualitative or
quantitative)

Figure out your goal
– Compare groups
– Measure relationship or association of
variables
Scales of Measurement

Nominal

Ordinal

Interval

Ratio
} Qualitative
} Quantitative
Nominal Scale (discrete)

Simplest scale of measurement

Variables which have no numerical value

Variables which have categories

Count number in each category, calculate
percentage

Examples:
–
–
–
–
–
Gender
Race
Marital status
Whether or not tumor recurred
Alive or dead
Ordinal Scale

Variables are in categories, but with an
underlying order to their values

Rank-order categories from highest to lowest

Intervals may not be equal

Count number in each category, calculate
percentage

Examples:
–
–
–
–
Cancer stages
Apgar scores
Pain ratings
Likert scale
Interval Scale

Quantitative data

Can add & subtract values

Cannot multiply & divide values
– No true zero point

Example:
– Temperature on a Celsius scale
• 00 indicates point when water will freeze, not an absence of
warmth
Ratio Scale (continuous)

Quantitative data with true zero
– Can add, subtract, multiply & divide

Examples:
–
–
–
–
–
Age
Body weight
Blood pressure
Length of hospital stay
Operating room time
Scales of Measurement

Nominal

Ordinal

Interval

Ratio
to nonparametric
} Lead
statistics
} Lead to parametric statistics
Two Branches of Statistics

Descriptive
– Frequencies & percents
– Measures of the middle
– Measures of variation

Inferential
– Nonparametric statistics
– Parametric statistics
Descriptive Statistics

First step in analyzing data

Goal is to communicate results, without
generalizing beyond sample to a larger
group
Frequencies and Percents
Number of times a specific value of an
observation occurs (counts)
 For each category, calculate percent of
sample

SMOKING
Valid
Missing
Total
smoker
non-smoker
Total
unknown
Frequency
26
79
105
22
127
Percent
20.5
62.2
82.7
17.3
100.0
Valid Percent
24.8
75.2
100.0
Cumulative
Percent
24.8
100.0
Measures of the Middle or
Central Tendency

Mean
– Average score
• sum of all values, divided by number of values
– Most common measure, but easily influenced by
outliers

Median
– 50th percentile score
• half above, half below
– Use when data are asymmetrical or skewed
Measures of Variation or Dispersion

Standard deviation (SD)
– Square root of the sum of squared deviations of the
values from the mean divided by the number of
values
SD =
sum of (individual value – mean value) 2
________________________________________________
number of values

Standard error (SE)
– Standard deviation divided by the square root of the
number of values
Measures of Variation or Dispersion

Variance
– Square of the standard deviation

Range
– Difference between the largest & smallest
value
nocigs_b
Valid
Missing
Total
1
2
3
5
6
12
13
14
15
17
18
19
20
22
24
30
39
40
45
100
Total
System
Frequency
2
1
1
3
1
1
1
1
2
1
1
2
2
1
1
1
1
1
1
1
26
101
127
Percent
1.6
.8
.8
2.4
.8
.8
.8
.8
1.6
.8
.8
1.6
1.6
.8
.8
.8
.8
.8
.8
.8
20.5
79.5
100.0
Valid Percent
7.7
3.8
3.8
11.5
3.8
3.8
3.8
3.8
7.7
3.8
3.8
7.7
7.7
3.8
3.8
3.8
3.8
3.8
3.8
3.8
100.0
Cumulative
Percent
7.7
11.5
15.4
26.9
30.8
34.6
38.5
42.3
50.0
53.8
57.7
65.4
73.1
76.9
80.8
84.6
88.5
92.3
96.2
100.0
Statistics
nocigs_b
N
Mean
Std. Error of Mean
Median
Mode
Std. Deviation
Variance
Range
Minimum
Maximum
Valid
Missing
26
101
19.62
3.985
16.00
5
20.320
412.886
99
1
100
Inferential Statistics
Sample

Population
Nonparametric tests
– Used for analyzing nominal & ordinal variables
– Makes no assumptions about data

Parametric tests
– Used for analyzing interval & ratio variables
– Makes assumptions about data
• Normal distribution
• Homogeneity of variance
• Independent observations
Which Test Do I Use?

Step 1
Know the scale of measurement

Step 2
Know your goal
– Is it to compare groups? How many groups
do I have?
– Is it to measure a relationship or association
between variables?
Key Inferential Statistics

Chi-Square
– Fisher’s exact test

T-test
– Unpaired
– Paired
}
}
Nonparametric
Association/Relationship
Parametric
Compare groups

Analysis of Variance (ANOVA)

Pearson’s Correlation

Linear Regression
}
}
Parametric
Compare groups
Parametric
Association/Relationship
Probability and p Values

p < 0.05
– 1 in 20 or 5% chance groups are not different
when we say groups are significantly different

p < 0.01
– 1 in 100 or 1% chance of error

p < 0.001
– 1 in 1000 or .1% chance of error
Research Hypothesis

Topic

Research question
–
hypothesis
Null hypothesis (H0)
•
–
research question
Predicts no effect or difference
Alternative hypothesis (H1)
•
Predicts an effect or difference
Example
Topic: Cancer & Smoking
Research Question: Is there a
relationship between smoking &
cancer?
H0: Smokers are not more likely to
develop cancer compared to nonsmokers.
H1: Smokers are more likely to
develop cancer than are non-smokers.
Are These Categorical
Variables Associated?
SMOKING * SES Crosstabulation
low
SMOKING
smoker
non-smoker
Total
Count
% within SES
Count
% within SES
Count
% within SES
7
38.9%
11
61.1%
18
100.0%
SES
middle
13
20.3%
51
79.7%
64
100.0%
high
Total
6
26.1%
17
73.9%
23
100.0%
26
24.8%
79
75.2%
105
100.0%
2
Chi-Square

Most common nonparametric test

Use to test for association between
categorical variables

Use to test the difference between observed
& expected proportions
– The larger the chi-square value, the more the
numbers in the table differ from those we would
expect if there were no association

Limitation
– Expected values must be equal to or larger than 5
Let’s Test For Association
Low SES 38.9%, Middle SES 20.3%, High SES 26.1%
Chi-Square Tests
Pearson Chi-Square
Likelihood Ratio
Linear-by-Linear
Association
N of Valid Cases
Value
2.630a
2.476
.653
2
2
Asymp. Sig.
(2-sided)
.268
.290
1
.419
df
105
a. 1 cells (16.7%) have expected count less than 5. The
minimum expected count is 4.46.
Alternative to Chi-Square

Fisher’s exact test
– Is based on exact probabilities
– Use when expected count <5 cases in
each cell and
– Use with 2 x 2 contingency table
R A Fisher 1890-1962
LUNG_CA * SMOKING Crosstabulation
LUNG_CA
positive
negative
Total
Count
% within SMOKING
Count
% within SMOKING
Count
% within SMOKING
SMOKING
smoker
non-smoker
3
1
11.5%
1.3%
23
78
88.5%
98.7%
26
79
100.0%
100.0%
Total
4
3.8%
101
96.2%
105
100.0%
Chi-Square Tests
Pearson Chi-Square
Continuity Correction a
Likelihood Ratio
Fisher's Exact Test
Linear-by-Linear
Association
N of Valid Cases
Value
5.633b
3.179
4.664
5.580
df
1
1
1
1
Asymp. Sig.
(2-sided)
.018
.075
.031
Exact Sig.
(2-sided)
Exact Sig.
(1-sided)
.046
.046
.018
105
a. Computed only for a 2x2 table
b. 2 cells (50.0%) have expected count less than 5. The minimum expected count is
.99.
Do These Groups Differ?
Group Statistics
BMI
SMOKING
smoker
non-smoker
N
26
79
Mean
25.1846
26.2228
Std. Deviation
5.27209
5.47664
Std. Error
Mean
1.03394
.61617
Unpaired t-test
or Student’s t-test
William Gossett 1876-1937

Frequently used statistical test

Use when there are two independent
groups
Unpaired t-test or Student’s
t-test

Test for a difference between groups
– Is the difference in sample means due to their
natural variability or to a real difference between
the groups in the population?

Outcome (dependent variable) is interval or
ratio

Assumptions of normality, homogeneity of
variance & independence of observations
Let’s Test For A Difference
Smokers’ BMI = 25.18 ± 5.27
Non-Smokers’ BMI = 26.22 ± 5.48
Independent Samples Test
Levene's Test for
Equality of Variances
F
BMI
Equal variances
assumed
Equal variances
not assumed
Sig .
.079
.779
t-test for Equality of Means
t
df
Sig . (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-.846
103
.400
-1.0382
1.22719
-3.47200
1.39566
-.863
44.127
.393
-1.0382
1.20362
-3.46371
1.38737
Do These Groups Differ?
Light smoker < 1 pack/day
Heavy smoker > 1 pack/day
Descriptives
BMI
N
non-smoker
light smoker
heavy smoker
Total
79
17
9
105
Mean
26.2228
26.1765
23.3111
25.9657
Std. Deviation
5.47664
4.96154
5.62015
5.42028
Std. Error
.61617
1.20335
1.87338
.52896
95% Confidence Interval for
Mean
Lower Bound
Upper Bound
24.9961
27.4495
23.6255
28.7275
18.9911
27.6311
24.9168
27.0147
Minimum
17.70
18.90
17.90
17.70
Maximum
40.20
35.00
35.90
40.20
Analysis of Variance (ANOVA)
or F-test

Three or more independent groups

Test for a difference between groups
– Is the difference in sample means due to their
natural variability or to a real difference between
the groups in the population?

Outcome (dependent variable) is interval or
ratio

Assumptions of normality, homogeneity of
variance & independence of observations
Let’s Test For A Difference
Non-Smokers’ BMI = 26.22 ± 5.48
Light Smokers’ BMI = 26.18 ± 4.96
Heavy Smokers’ BMI = 23.31 ± 5.62
ANOVA
BMI
Between Groups
Within Groups
Total
Sum of
Squares
69.398
2986.058
3055.457
df
2
102
104
Mean Square
34.699
29.275
F
1.185
Sig .
.310
Is there a
relationship
between the
variables?
No_Cigs
1
1
2
3
5
5
5
6
12
13
14
15
15
17
18
19
19
20
20
22
24
30
39
40
45
100
BMI
30.1
18.9
22.8
22.6
24.2
26.2
33.3
19.1
35
23
22.2
28.7
28.6
24.3
30.9
22.5
32.6
19
26.7
18.8
23.4
23.2
25
35.9
17.9
19.9
Pearson’s Correlation
Karl Pearson 1857-1936

Measures the degree of relationship between
two variables

Assumptions:
– Variables are normally distributed
– Relationship is linear
– Both variables are measured on the interval or
ratio scale
– Variables are measured on the same subjects
Scatterplots
Perfect positive
correlation
r = -1.0 ---- +1.0
Perfect negative
correlation
No correlation
Let’s Test For A Relationship
Correlations
NOCIGS_B
BMI
Pearson Correlation
Sig . (2-tailed)
N
Pearson Correlation
Sig . (2-tailed)
N
NOCIGS_B
1
.
26
-.169
.410
26
BMI
-.169
.410
26
1
.
105
40
30
BMI
20
10
0
20
NOCIGS_B
40
60
80
100
120
Interpretation of Results

The size of the p value does not
indicate the importance of the result

Appropriate interpretation of statistical
test
– Group differences
– Association or relationship
– “Correlation does not imply causation”
Don’t Lie With Statistics !