Testing of Hypothesis
D.A. Asir John Samuel, BSc (Psy), MPT (Neuro Paed), MAc, DYScEd,
C/BLS, FAGE
Hypothesis
• Hypothesis is defined as the statement
regarding parameter (characteristic of a
population)
Dr.Asir John Samuel (PT), Lecturer, ACP 2
Test of significance
• A statistical procedure by which one can
conclude, if the observed results from the
sample is due to chance (sampling variation)
or not
Dr.Asir John Samuel (PT), Lecturer, ACP 3
A B
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
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Null hypothesis (H0)
• A hypothesis which states that the observed
result is due to chance
• Researcher anticipate “no difference” or “no
relationship”
Dr.Asir John Samuel (PT), Lecturer, ACP 5
Alternate hypothesis (HA)
• A hypothesis which states that the observed
results is not due to chance (research
hypothesis)
• Statement predict that a difference or
relationship b/w groups will be demonstrated
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Testing of hypothesis
1. Evaluate data
2. Review assumption
3. State hypothesis
4. Presume null hypothesis
5. Select test statistics
6. Determine distribution of test statistics
7. State decision rule Dr.Asir John Samuel (PT), Lecturer, ACP 7
Testing of hypothesis
8. Calculate test statistics
9. What is the probability that the data conform
10. Make statistical decision
11. If p>0.05, then reject (HA)
12. If p<0.05, then accept (HA)
Dr.Asir John Samuel (PT), Lecturer, ACP 8
Testing of Hypothesis
Presume null hypothesis
What is the probability
that data conform to
null hypothesis
Retain H0 reject H0
p>0.05 P<0.05
Dr.Asir John Samuel (PT), Lecturer, ACP 9
p-value
• Probability of getting a minimal difference of
what has observed is due to chance
• Probability that the difference of at least as
large as those found in the data would have
occurred by chance
Dr.Asir John Samuel (PT), Lecturer, ACP 10
p value in decision
• P value very large
- Probability to get the observed result due to
chance
• P value very small
- Small probability to get the observed result
not due to chance
Dr.Asir John Samuel (PT), Lecturer, ACP 11
Decision for 5% LOS
• Probability of rejecting true null hypothesis
• If p-value <0.05, then data favours alternate
hypothesis
• If p-value ≥0.05, then data favours null
hypothesis
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Type I & II errors
Possible states of Null Hypothesis
Possible actions on
Null Hypothesis
True False
Accept Correct Action
Type II error
Reject Type I error
Correct Action
Prob (Type I error) – α (LoS) Prob (Type II error) – β 1-β – power of test
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LOS and Power
• Prob (type I error) = α
• Prob (type II error) = β
• α – LOS
• 1- β – power of the study
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Test of Hypothesis
• Parametric test
• Non-parametric test
Dr.Asir John Samuel (PT), Lecturer, ACP 15
Parametric & non-parametric test
• Paired t-test
• Repeated measure
ANOVA
• Independent t-test
• One way ANOVA
• Pearson correlation
coefficient
• Wilcoxon Signed Rank T
• Friedman test
• Mann-Whitney U test
• Krushal Wallis test
• Spearman Rank
correlation coefficient Dr.Asir John Samuel (PT), Lecturer, ACP 16
Paired t-test
• Two measures taken on the same subject or
naturally occurring pairs of observation or two
individually matched samples
• Variable of interest is quantitative
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Assumption
• The difference b/w pairs in the population is
independent and normally or approximately
normally distributed
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Wilcoxon Signed Rank test
• Used for paired data
• The sample is random
• The variable of interest is continuous
• The measurement scale is at least interval
• Based on the rank of difference of each paired
values Dr.Asir John Samuel (PT), Lecturer, ACP 19
Repeated measures ANOVA
• Measurements of the same variable are made
on each subject on more than two different
occasion
• The different occasions may be different point
of time or different conditions or different
treatments
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Assumptions
• Observations are independent
• Differences should follow normal distribution
• Sphericity-differences have approximately
same variances
Dr.Asir John Samuel (PT), Lecturer, ACP 21
Fried Man test
• Data is measured in ordinal scale
• The subjects are repeatedly observed under 3
or more conditions
• The measurement scale is at least ordinal
(qualitative)
• The variable of interest is continuous Dr.Asir John Samuel (PT), Lecturer, ACP 22
Independent t-test
• Compare the means of two independent
random samples from two population
• Variable of interest is quantitative
Dr.Asir John Samuel (PT), Lecturer, ACP 23
Assumptions
• The population from which the sample were
obtained must be normally or approximately
normally distributed
• The variances of the population must be equal
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Mann Whitney-U test
• Two independent samples have been drawn
from population with equal medians
• Samples are selected independently and at
random
• Population differ only with respect to their
median
• Variable of interval is continuous Dr.Asir John Samuel (PT), Lecturer, ACP 25
Mann Whitney-U test
• Measurement scale is at least ordinal
(qualitative)
• Based on ranks of the observations
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ANOVA
• Extension of independent t-test to compare
the means of more than two groups
• F = b/w group variation/within group variation
• F ratio
• Post hoc test (which mean is different)
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Assumptions
• Observations are independent and randomly
selected
• Each group data follows normal distribution
• All groups are equally variable (homogeneity
of variance)
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Why not t-test?
• Tedious
• Time consuming
• Confusing
• Potentially misleading – Type I error is more
Dr.Asir John Samuel (PT), Lecturer, ACP 29
Kruskal Wallis H test
• Used for comparison of more than 2 groups
• Extension of Mann-Whitney U test
• Used for comparing medians of more than 2
groups
Dr.Asir John Samuel (PT), Lecturer, ACP 30
Assumptions
• Samples are independent and randomly
selected
• Measurement scale is at least ordinal
• Variable of interest is continuous
• Population differ only with respect to their
medians Dr.Asir John Samuel (PT), Lecturer, ACP 31
Chi-square Test (x2)
• Variables of interest are categorical
(quantitative)
• To determine whether observed difference in
proportion b/w the study groups are
statistically significant
• To test association of 2 variables
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Chi-square Test-Assumption
• Randomly drawn sample
• Data must be reported in number
• Observed frequency should not be too small
• When observed frequency is too small and
corresponding expected frequency is less than
5 (<5) – Fischer Exact test Dr.Asir John Samuel (PT), Lecturer, ACP 33
Relationship
• Correlation
• Regression
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Correlation
• Method of analysis to use when studying the
possible association b/w two continuous
variables
• E.g.
- Birth wt and gestational period
- Anatomical dead space and ht
- Plasma volume and body weight
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Correlation
• Scatter diagram
• Linear correlation
• Non-linear correlation
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Properties
• Scatter diagrams are used to demonstrate the
linear relationship b/w two quantitative
variables
• Pearson’s correlation coefficient is denoted by r
• r measures the strength of linear relationship
b/w two continuous variable (say x and y)
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Properties
• The sign of the correlation coefficient tells us
the direction of linear relationship
• The size (magnitude) of the correlation
coefficient r tells us the strength of a linear
relationship
Dr.Asir John Samuel (PT), Lecturer, ACP 38
Properties
• Better the points on the scatter diagram
approximate a straight line, the greater is the
magnitude r
• Coefficient ranges from -1 ≤ r ≤ 1
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Interpretation
• r = 1, two variables have a perfect +ve linear
relationship
• r = -1, two variables have a perfect -ve linear
relationship
• r = 0, there is no linear relationship b/w two
variables
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Assumption
• Observations are independent
• Relationship b/w two variables are linear
• Both variables should be normal distributed
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Caution
• Correlation coefficient only gives us an
indication about the strength of a linear
relationship
• Two variables may have a strong curvilinear
relationship but they could have a weak value
for r
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Judging the strength – Porteney & Watkins criteria
• 0.00-0.25 – little or no relationship
• 0.26-0.50 – fair degree of relationship
• 0.51-0.75 – moderate to good degree of
relationship
• 0.76-1.00 – good to excellent relationship
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Scatter diagram
• Each pair of variables is represented in scatter
diagram by a dot located at the point (x,y)
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Scatter diagram - Merits
• Simple method
• Easy to understand
• Uninfluenced
• First step
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Scatter diagram - Demerits
• Does not establish exact degree of correlation
• Qualitative method
• Not suitable for large sample
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Spearman’s Rank correlation
• Non-parametric measure of correlation
between the two variables (at least ordinal)
• Ranges from -1 to +1
Eg:
- Pain score of age
- IQ and TV watched /wk
- Age and EEG output values Dr.Asir John Samuel (PT), Lecturer, ACP 47
Situation
• Relationship b/w two variables is non-linear
• Variables measured are at least ordinal
• One of the variables not following normal
distribution
• Based on the difference in rank between each
variable
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Assumption
• Observation are independent
• Samples are randomly selected
• The measurement scale is at least ordinal
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Regression
• Expresses the linear relationship in the form of
an equation
• In other words a prediction equation for
estimating the values of one variable given the
valve of the other,
y = a + bx
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Regression - eg
• Wt (y) and ht (x)
• Birth wt (y) and gestation period (x)
• Dead space (y) and height (x)
x and y are continuous
y-dependent variable
x-independent variable Dr.Asir John Samuel (PT), Lecturer, ACP 51
Regression line
• Shows how are variable changes on average
with another
• It can be used to find out what one variable is
likely to be (predict) when we know the other
provided the prediction is within the limits of
data range
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Regression analysis
• Derives a prediction equation for estimating
the value of one variable (dependent) given the
value of the second variable (independent)
y = a + bx
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Assumption
• Randomly selection
• Linear relationship between variables
• The response variable should have a normal
distribution
• The variability of y should be the same for
each value of the predictor value
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Multiple regression
• One dependent variable and multiple
independent variable
• Derives a prediction equation for estimating
the value of one variable (dependent) given
the variable of the other variable
(independent)
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Multiple regression
• The dependent variable is continuous and
follows normal distribution
• Independent variable can be quantitative as
well as qualitative
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