Inferential statistics
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In inferential statistics
• Data from samples are used to make
inferences about populations
• Researchers can make generalizations
about an entire population based on a
smaller number of observations
• However, the sample means will not all be
the same when repeated random samples
are taken from a population
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Sampling distributions
• If many different samples were taken from
a population, it would produce a
distribution of sample means
• If repeated enough times, the distribution
would take on a normal shape
– Even if the underlying population is not
normal
• If repeated an infinite number of times, it
would be called a sampling distribution
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Sampling distributions (cont.)
• Which of the sample means is truly the
population mean?
– It would be useful to know, but an exact figure
is not possible
• The population mean can be inferred from
the sample
– The sample mean is an estimate
– Referred to as the point estimate
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Sampling distributions (cont.)
• Because sampling distributions are
normal, the properties of the normal
distribution can be used
– e.g., the 68.3, 95.5, 99.7 proportion of the
area under the curve
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Standard error of the mean
(SEm)
• The spread of means around the mean of
a sampling distribution
• Can be estimated from the sample
– SEm is calculated by dividing the SD of the
sample by the square root of the number of
units in the sample
SE m  S
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SEm (cont.)
• SEm is higher when
– The sample’s SD is large or
– The sample size is small
• Lower when
– SD is a small or
– The sample size is large
• A small SEm is preferable because
generalizations are more precise
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Confidence Intervals (CIs)
• A CI is a range of values that is likely to
contain the population parameter that is
being estimated (e.g., the mean)
• The probability that this range of values
contains the population parameter is
typically 95%
– Thus, the 95% confidence interval
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Confidence Intervals (CIs)
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-2
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0
+1
9
+2
+3
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CIs (cont.)
•
One can have 95% confidence that the
value of the true mean lies within the
calculated interval (i.e., 95% CI)
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Calculating a 95% CI
1. Find the z-score (using a z-table) that
corresponds to the area under the
distribution that includes 95% of all
values (e.g., z = ±1.96 for a 95% CI)
2. Multiply the z-scores by the SEm
3. Add the product to the sample mean to
find the upper limit of the CI and subtract
to find the lower limit
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Size (width) of CIs
• The size of the CI is related to the size of
the sample and the size of the data
variation
– Small samples & large variation = larger CIs
– Large samples & small variation = smaller CIs
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Hypothesis testing
• A hypothesis is an assumption that
appears to explain certain events, which
must be tested to see whether it is true
• Research hypothesis
– a.k.a., alternative hypothesis
– Denoted H1
– The research hypothesis is not tested directly
• Instead the null hypothesis (H0) is tested
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Hypothesis tests
• Depending on the outcome of the test of
H0, there is either support for or against
the research hypothesis
• Hypothesis testing involves the
comparison of the means of groups in an
experiment
– The objective is to find out whether they are
significantly different from each other
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Hypothesis tests (cont.)
• When comparing the means of an active
treatment group and a control group, one
looks for a difference
– The treatment may produce a better outcome
leading to a higher mean than the control
group
– The difference may appear real, but it may be
due to chance
– Statistical tests verify if it is real
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The null hypothesis
• H0 states that there is no difference
between the group means
• H1 is accepted only if the null hypothesis
proves to be unlikely
– Typically it must be at least 95% unlikely
– If H0 is unlikely, it is rejected
• Not unlike the innocent until proven guilty
concept in our legal system
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A hypothetical neck pain study
• Patients are treated with chiropractic vs.
usual medical care
– Outcome measure is the Neck Disability Index
(NDI)
– H1
• Chiropractic patients will have lower mean NDI
scores after treatment
– H0
• There is no difference between mean NDI scores
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Hypothetical study (cont.)
• Results
– Mean NDI scores of chiropractic patients
• 28 before, 10 after treatment
– Mean NDI scores medical patients
• 29 before, 15 after treatment
• Chiropractic care appears to be better
– But is there enough difference to rule out
chance
– Must perform statistical tests to find out
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Hypothetical study (cont.)
Chiropractic
Medical
NDI score
30
Is this difference
enough to be
meaningful?
20
10
0
Baseline
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Outcome
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Statistical significance
• The results of a study (i.e., the difference
between groups) are unlikely to be due to
chance
– At a specified probability level, referred to as
alpha ()
–  is the probability of incorrectly rejecting a
null hypothesis
• If the results are not due to chance, H0 is
rejected and H1 is accepted
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Statistical significance (cont.)
• It must be at least 95% unlikely that H0 is
true before it can be rejected
• There is still a 5% chance that H0 would
be rejected, when it was actually true
• Accordingly, P values must be equal to or
less than 5% in order for the results of a
study to reach a level of statistical
significance
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Statistical significance (cont.)
• The level of significance (alpha level) is
not the same as the P value
– The alpha level must be set before the study
begins
– The P value is calculated at the completion of
the study and must be ≤ to the alpha level in
order to reach statistical significance
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Statistical significance (cont.)
• Even when studies are not statistically
significant, there is a 1:20 chance that
significant results would occur if the study
was repeated 20 times
• Fishing
– When researchers perform a lot of statistical
tests on their data
– Increases the chance that at least one of the
tests will wrongly reach statistical significance
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Type I & II errors
• Type I error (a.k.a., alpha error)
– Rejecting a true null hypothesis
– The probability of making a Type I error is
equal to the value of α
• Type II error (a.k.a., beta error )
– Failure to reject a false null hypothesis
– The probability of making a Type II error is
equal to the value of beta ()
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Type I & II errors (cont.)
Consequences of accepting or rejecting
true and false null hypotheses
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Type I & II errors (cont.)
• There is a trade-off between the likelihood
of a study resulting in a Type I error versus
a Type II error
• As alpha becomes smaller, the chance of
making a Type I error decreases
• Whereas the chance of making a Type II
error increases
– Because it is more likely that a false H0 will
not be rejected
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Type I & II errors (cont.)
The 0.05 alpha level
is a compromise
between Type I and
Type II errors
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Power
• The probability of correctly rejecting a
false H0
– Related to  error
– Power is equal to 1-
• Power depends on sample size, the
magnitude of the difference between
group means, and the value of α
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Power (cont.)
• Power increases as
– Sample size increases
• Only to a certain extent, then it becomes a waste
of resources
– The difference between group means
increases
– α increases
• A power value of 0.80 is often sought by
researchers
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Power (cont.)
• Power may be calculated after a study has
been completed (post hoc)
– If low power is detected during post hoc
power analysis and H0 was not rejected, it
may be grounds to repeat the study using a
larger sample
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Confidence intervals and
hypothesis testing
• If the value specified as the difference
between group means in the null
hypothesis is included in the 95% CI, then
H0 should not be rejected
– The test is not statistically significant
• H0 states there is no difference between
group means, so the specified no
difference value is always zero
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CIs and hypothesis
testing (cont.)
• If zero is not included in the 95% CI, the
null hypothesis should be rejected
– The test is statistically significant
• CIs are becoming more prevalent in the
health care literature because they convey
more information than P values alone
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CIs and hypothesis
testing (cont.)
• Example study
– Brinkhaus et al.
– Acupuncture was more effective in improving
pain on VAS* than no acupuncture in chronic
low back pain patients
• Difference, 21.7 mm (95% CI 13.9 to 30.0)
– But no statistical difference between
acupuncture and minimal acupuncture
• Difference, 5.1 mm (95% CI -3.7 to 13.9)
* Visual analog scale
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Clinical significance
a.k.a., practical significance
• Do the findings of a study really matter in
clinical situations
• Sometimes a study is statistically
significant, but the findings are not
important in clinical terms
– Large studies with small differences between
groups can generate statistically significant
findings that are not meaningful to
practitioners
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Clinical significance (cont.)
• For example
– A study found a statistically significant
difference between mean Headache Disability
Inventory (HDI) scores of only 10 points
– Yet at least a 29-point change must occur
from test to retest before the changes can be
attributed to a patient’s treatment
• The HDI is not very responsive to change
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Commonly encountered
statistical tests
• Statistical tests determine the probabilities
associated with relationships in studies
– Are the results real or merely due to chance?
• t-test, ANOVA, and chi-square are
common in journal articles
– Familiarity with these tests is helpful in the
appraisal of articles
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t-test
• Used to find out whether the means of two
groups are statistically different
• Results are not entirely black-and-white
– Only indicates that the means are probably
different
– Or, that they are probably the same, if the
study fails to find a difference
• The t-test can be used for a single group
by comparing the mean with known values
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t-test (cont.)
• The actual differences between means is
considered
• Also the amount of variability of the scores
– A high degree of variability of group scores
can obscure the differences between means
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t-test (cont.)
• The differences between
means are the same in both
examples, but the variability
of group scores differs
• The lower example would
be much more likely to
reach statistical significance
because of the narrow
spread
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Assumptions of the t-test
• The data should be normal and involve
interval or ratio measurement
• Groups should be independent
• The variances of groups should be equal
• When the sample size is large enough
(about 30 subjects) violations of these
assumptions are less important
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Alternatives to the t-test
• The t-test for unequal variances
• Non-parametric tests for use with skewed
data
– Mann-Whitney U test
– Wilcoxon test
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The t-score
• The t-score (a.k.a., t-ratio) is similar to the
z-score
– However, the t-distribution and a t-table are
used
– This is because the SD of the population is
estimated from the sample, whereas it is
known in the z-distribution
• P values are found using the calculated
t-score and a t-table
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The t-score (cont.)
• t-tables consider the number of subjects in
the groups
• Referred to as degrees of freedom (df)
– Signifies the number of subjects in each
group minus 1
– Minus 2 when there are two groups
– Thus, a study that compares the means of 2
groups that involve 30 subjects has 28 df
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The t-table
• t-distributions eventually become nearly
normal when many subjects are included
– As a result, t-tables usually only go to 100 df
• Alpha levels are shown for
– When α is all in one tail (α1 or one-tailed test )
– When α is spit between the tails (α2 or twotailed test)
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t-table showing critical values for t.
(only to 15 df)
df
1
0.10
0.05
0.025
0.01
0.005
0.001
0.0005
2
0.20
0.10
0.05
0.02
0.01
0.002
0.001
1
3.078
2
1.886
3
1.638
4
1.533
5
1.476
6
1.440
7
1.415
8
1.397
9
1.383
10
1.372
11
1.363
12
1.356
13
1.350
14
1.345
15
1.341
…
To 100
Etc.Evidence-based
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6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
12.71
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
31.82
63.66
318.30
6.965
9.925
22.330
4.541
5.841
10.210
3.747
4.604
7.173
3.365
4.032
5.893
3.143
3.707
5.208
2.998
3.499
4.785
2.896
3.355 for 10
4.501
Critical value
df
2.821
4.297
and α2 =3.250
0.05
2.764
3.169
4.144
2.718
3.106
4.025
2.681
3.055
3.930
2.650
3.012
3.852
2.624
2.977
3.787
2.602
2.947
3.733
45
636.62
31.60
12.92
8.610
6.869
5.959
5.408
5.041
4.781
4.587
4.437
4.318
4.221
4.140
4.073
© 2006
One-tailed test vs.
two-tailed test
• One-tailed test (a.k.a., directional test)
– Alpha is all in one tail
– The researcher specifies the direction the test
results will go before the data analysis
• Either higher or lower
• Two-tailed test (a.k.a., non-directional test)
– Alpha is split between the tails
– The study’s results could go either way
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One-tailed test vs.
two-tailed test (cont.)
• In a non-directional test, the researcher
wants to know if the means are different
– For example, in a study comparing
manipulation with acupuncture for tension
headaches, the results could go either way
– That is the case with almost all studies that
compare treatments
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One-tailed test vs.
two-tailed test (cont.)
• It is easier to reach statistical significance
using a directional test
– Consequently it is tempting for researchers to
use directional hypotheses
• The opposite direction must be of no
interest to the researcher
– But it is almost always possible for the test to
go either way when comparing treatments
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Calculating the t-score
• Is a ratio of the difference between group
means and the variability of the data
• Variability is represented by the standard
error of the difference ( SX X ) rather than
the SD
• Thus
1
t
X1  X 2
S X1  X 2
Evidence-based Chiropractic
or
t
2
T hedifferencebetween group means
Variability of thedata
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The t-score
• For the t-test result to be statistically
significant
– The difference between the means must be
large (the numerator)
– And the variability of the data must be small
(the denominator)
• This results in a t-score that is larger than
the critical value of t in the t-table
Evidence-based Chiropractic
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The t-score (cont.)
• Remember
Big t-value
Small P value
Statistical significance
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Steps involved in the t-test
• Calculate the means and standard deviations of
the groups’ outcomes
• Calculate the t-ratio
• Check to see if the calculated t is statistically
significant using a t-table
• It is significant if t is greater than the critical
value of t at the 0.05 level
• If so, the group means are considered different
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Reporting t-test
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Paired t-test
• Groups are dependent
– The same subjects are in each of the groups
• e.g., repeated measures studies
– Or subjects are matched
• e.g., twins or when subjects are very much alike
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Analysis of variance (ANOVA)
• Used to compare means when more than
two groups are involved
• Repeating t-tests increases the probability
of producing a Type I error
• ANOVA can only compare one outcome
variable
– Univariate
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ANOVA (cont.)
•
ANOVA provides information about
– Whether there are any significant
differences among the group means
– Whether any of the particular groups differ
from each other
– Whether the differences are relatively big or
small
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The F-ratio
•
Not unlike the t-ratio, the F-test
compares the variance between the
groups with the variance within the
groups
F=
Evidence-based Chiropractic
variance between groups
variance within the groups
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The F-ratio (cont.)
• Within-group variance is related to
sampling error and ordinary differences
between subjects
– For instance, many physical characteristics
vary normally (e.g., cortisol levels, pulse rate,
and blood pressure)
• Between-group variance is related to the
differences between the means
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Between and
within-group variance
The means of
3 groups are
compared
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The F-ratio (cont.)
•
•
•
If the F-ratio is small, the groups are
probably not significantly different
If it is big, at least two of the groups are
significantly different
The F- test does not identify which of the
groups are different
– Comparison tests are necessary
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Comparison tests
• Compare the group pairs (pairwise)
• Common comparison tests include
– Tukey
• Used if the groups are of unequal size
– Bonferroni
• For both equal and unequal group sizes
– Scheffé
• Is very conservative to minimize the risk of type I
error
Evidence-based Chiropractic
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Microsoft Excel has several
built-in statistical functions
• Excel can be used to calculate ANOVA,
t-ratio, and others
• Select Data Analysis from the Tools menu
and input the data
Evidence-based Chiropractic
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ANOVA table
Anova: Single Factor
SUMMARY
Groups
Count
Sum
Mean
Variance
Chiropractic care
8
109
13.625
8. 55357
Medical care
8
54
6.75
7.92857
PT care
8
51
6.375
11.125
• There is not much difference between the means of the
medical and PT groups (6.75 and 6.38)
• But the mean of the chiropractic group (13.63) appears
to be different
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ANOVA table (cont.)
ANOVA
Source of
Variation
Between Groups
Within Groups
Total
Sum of
squares
SS
266.5833
df
MS
2 133.2917
193.25
21
459.8333
23
Mean
squares
F
14.4845
P value
F crit
0.00011
3.4668
9.2024
The critical
value of F
• The calculated value of F (14.5) exceeds the critical
value of F (3.5), so the group means are different overall
(P < 0.001)
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Comparison test results
Tukey HSD
(I) Type of (J) Type of
care
care
Chiro
MD
Std.
Error
P
value
95% Confidence
Interval
MD
6.87500*
1.51677
.001
3.0519 to 10.6981
PT
7.25000*
1.51677
.000
3.4269 to 11.0731
-6.87500*
1.51677
.001
-10.6981 to -3.0519
.37500
1.51677
.967
-3.4481 to 4.1981
-7.25000*
1.51677
.000
-11.0731 to -3.4269
-.37500
1.51677
.967
-4.1981 to 3.4481
Chiro
PT
PT
Difference
(I-J)
Chiro
MD
* The mean difference is significant at the .05 level.
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Assumptions of ANOVA test
•
•
•
•
Normally distributed data
Groups should be independent
Variances of groups should be equal
If not, a nonparametric test should be used
– Kruskal-Wallis test
• When variances are unequal
– Friedman test
• When paired groups are involved
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Chi-square test
• Used to test hypotheses involving
categorical data
• There are 2 versions
– Chi-square goodness of fit
• Determines if observed frequencies of occurrence
differ from what would be expected by chance
– Chi-square test of independence
• Tests to see if frequencies for one category differ
significantly from those of another category
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Chi-square goodness of fit
• Called the goodness of fit test because it
tests whether observed frequencies “fit”
against the expected frequencies
• For example
– If a sample of Americans found 60 males and
40 females, would that be statistically
significantly different from what would
normally be expected (50/50)?
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Goodness of fit example (cont.)
2
• Chi-square (Χ ) calculates the difference
between the observed and expected
frequencies, then divides that value by the
2
expected frequencies to generate the Χ statistic
Χ = Σ(O-E)
E
2
2
O – observed frequencies
E – expected frequencies
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Goodness of fit example (cont.)
• Following are the calculations for the 100
Americans example
Observed
Χ = (60 – 50)
50
2
Expected
2
+
(40 – 50)
50
2
2
Χ = 4.0
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Goodness of fit example (cont.)
• A chi-square table is used to see if the
results are statistically significant
– Only if the critical value is exceeded (3.84 in
this case)
• df is the number of categories minus 1
2
• The calculated Χ is 4
– So, the sample is different from what was
expected
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Chi-square test
of independence
• Frequencies of one variable are compared
with another to see if they differ
significantly
• A 2 X 2 contingency table (a.k.a., crosstabulation table) is used
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A 2 X 2 contingency table
Variable 1
Variable 2
Yes
No
Yes
a
b
a+b
No
c
d
c+d
a+c
b+d
Column Total
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Row Total
a+b+c+d
Grand Total
© 2006
Example hypothetical study
• Two groups of patients are treated using
different spinal manipulation techniques
– Gonstead vs. Diversified
• The presence or absence of pain after
treatment is the outcome measure
• Two categories
– Technique used
– Pain after treatment
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Gonstead vs. Diversified
example - Results
Technique
Pain after treatment
Yes
No
Row Total
Gonstead
9
21
30
Diversified
11
29
40
Column Total
20
50
70
Grand Total
9 out of 30 (30%) still had pain after Gonstead treatment
and 11 out of 40 (27.5%) still had pain after Diversified,
but is this difference statistically significant?
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Gonstead vs. Diversified
example (cont.)
2
• Calculating Χ
• First find the expected values for each cell
Expected (E) = Row total Χ Column total
Grand total
2
• Then calculate the Χ statistic using the cells’
expected (E) values and the previously provided
2
Χ formula
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Gonstead vs. Diversified
example (cont.)
• To find E for cell a (and similarly for the rest)
Technique
Pain after treatment
Yes
No
Row Total
Gonstead
9
21
30
Diversified
11
29
40
Column Total
20
50
70
Divide by grand total
Grand Total
Times column total
Evidence-based Chiropractic
Multiply row total
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Gonstead vs. Diversified
example (cont.)
• Find E for all cells
Technique
Pain after treatment
Yes
No
Row Total
Gonstead
9
E = 30*20/70=8.6
21
E = 30*50/70=21.4
Diversified
11
E=40*20/70=11.4
29
E=40*50/70=28.6
40
Column Total
20
50
70
30
Grand Total
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Gonstead vs. Diversified
example (cont.)
2
• Use the Χ formula with each cell and then
add them together
(9 - 8.6)2
8.6
11.4)2
(11 11.4
(21 - 21.4)2
21.4
28.6)2
(29 28.6
0.0186 0.0168
=
0.0316 0.0056
Χ2 = 0.0186 + 0.0168 + 0.0316 + 0.0056 = 0.0726
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Gonstead vs. Diversified
example (cont.)
2
• Find df and then consult a Χ table to see if
statistically significant
– df = (number of categories for variable 1) -1 X
(number of categories for variable 2) -1
• There are two categories for each variable
in this case, so df = 1
• Critical value at the 0.05 level and one df
is 3.84
2
– Therefore, Χ is not statistically significant
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Χ required conditions
2
• Observations must be independent
– The total number of observed frequencies
should not be higher than the number of
subjects in the study
• No small expected frequencies
– Expected frequencies less than one or less
than five in more than 20 percent of cells are
too small
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Χ requirements (cont.)
2
– Fisher's exact test
• An alternative to the chi-square test that is used
when expected frequencies are too small
• All that is needed is at least one data value in each
row and one data value in each column
• No extremely small or extremely large
samples
– Extremely small samples may overlook
obvious false null hypotheses and extremely
large samples may identify trivial differences
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Correlation
• A measure of mathematical relationships
that may exist between two or more
variables
– i.e., if one variable increases or decreases,
the other one will also increase or decrease a
specific amount
• Pearson’s correlation coefficient (r) is
commonly used
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Correlation (cont.)
• Correlation coefficient values range from
-1 to +1
+1 = perfect positive correlation
-1 = perfect negative correlation
• The closer r is to +1 or -1, the more closely
variables are related
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Correlation (cont.)
• Positive r values
– Variables tend to go up or down together
• Negative r values
– Variables tend to go up and down in
opposition
• An r value of 0
– There is no mathematical relationship
between variables
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Correlation (cont.)
• The units of measurement that are used
do not affect correlation coefficient
calculations
– e.g., height and weight results will be the
same whether in and lb or cm and kg are
used in the calculation
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No cause-and-effect
• A strong relationship between two
variables does not mean that one caused
the other to change
• For instance, there is a strong relationship
between coffee drinking and developing
lung cancer
– Actually, heavy coffee drinkers tend to be
heavy smokers
– Smoking is the actual cause
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Scatterplots
• An X-Y graph with symbols that represent
the values of two variables
Regression
line
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Examples
Positive correlation
slopes upward
Negative correlation
slopes downward
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Examples (cont.)
No correlation
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Scatterplots (cont.)
• Show the form, direction, and strength of
the relationship between variables
• Its form may be linear, but can also be
curvilinear or nonlinear
• A correlation weakens after a certain point
when data is curvilinear
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Curvilinear example
• As people age they get stronger to a certain
point, but as they continue to age, they
eventually begin to weaken
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Outliers
• Extreme values that are located far away
from the group of data on a scatterplot
• Outliers can strongly influence the slope of
the regression line
– And the value of the correlation coefficient
• Authors should adequately discuss outliers
– Why they occurred
– How they were dealt with
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Outliers (cont.)
• Outliers are obvious on a scatterplot
Outlier
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Coefficient of determination
• Is the correlation coefficient squared
– Symbolized as r2
• Only positive values are possible (because
it is squared)
– Ranging from 0 to 1
• Denotes how much of the variation in one
variable can be explained by the other
variable
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Coefficient of determination
• Example
– If a study on the relationship between the
amount lifted at work and the incidence of
low-back pain reported r2 as 0.65
– One could say that 65% of the variability in
the incidence of low-back pain was explained
by the amount workers lifted
– Other factors are responsible for the
remaining 35% variability
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Regression
• Regression analysis
– Calculation of the line of best fit passing
through a set of data
– An equation is generated that describes the
line of best fit (a.k.a., least squares line)
• Using the equation, predictions can be
made about the direction and amount
variables change
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Regression (cont.)
• A regression line is fitted by minimizing the
sum of squared deviations of the data
points from the least squares line
• The regression equation is Y = a + bX,
where
– a is the Y intercept
– b is the slope of the line
– X is the value of the (predictor) variable
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Regression (cont.)
Y
a
b
X
The value of Y can be calculated from a given
value of X
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The regression line
The line is positioned so that the distances of all
deviations are as short as possible
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Multiple regression
• Frequently outcomes are affected by more
than one predictor variable
• The multiple regression equation is similar
to simple regression, but with more than
one value for b. Thus, the equation is
Y = a + b1X1 + b2X2 + . . . + bkXk, where
• X1 is the first predictor variable, X2 is the second,
and Xk continues for as many predictor variables
as are involved
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