9.testing of hypothesis

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


A B
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10


Null hypothesis (H0)

• A hypothesis which states that the observed
result is due to chance

• Researcher anticipate “no difference” or “no
relationship”


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


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

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)


Testing of Hypothesis
Presume null hypothesis

What is the probability
that data conform to
p>0.05 null hypothesis P<0.05

Retain H0 reject H0

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


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

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


Type I & II errors
Possible states of Null Hypothesis
Possible True False
actions on Accept Correct Type II
Null Action error
Hypothesis Reject Type I Correct
error Action
Prob (Type I error) – α (LoS)
Prob (Type II error) – β
1-β – power of test

LOS and Power

• Prob (type I error) = α

• Prob (type II error) = β

• α – LOS

• 1- β – power of the study


Test of Hypothesis

• Parametric test

• Non-parametric test


Parametric & non-parametric test
• Paired t-test • Wilcoxon Signed Rank T
• Repeated measure • Friedman test
ANOVA
• Independent t-test • Mann-Whitney U test
• One way ANOVA • Krushal Wallis test
• Pearson correlation • Spearman Rank
coefficient correlation coefficient

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


Assumption

• The difference b/w pairs in the population is

independent and normally or approximately

normally distributed


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

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


Assumptions

• Observations are independent

• Differences should follow normal distribution

• Sphericity-differences have approximately
same variances


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

Independent t-test

• Compare the means of two independent

random samples from two population

• Variable of interest is quantitative


Assumptions

• The population from which the sample were
obtained must be normally or approximately
normally distributed

• The variances of the population must be equal


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

Mann Whitney-U test

• Measurement scale is at least ordinal
(qualitative)

• Based on ranks of the observations


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)


Assumptions

• Observations are independent and randomly
selected

• Each group data follows normal distribution

• All groups are equally variable (homogeneity
of variance)


Why not t-test?

• Tedious

• Time consuming

• Confusing

• Potentially misleading – Type I error is more


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


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

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

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

Relationship

• Correlation

• Regression


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

Correlation

• Scatter diagram

• Linear correlation

• Non-linear correlation


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)

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


Properties

• Better the points on the scatter diagram
approximate a straight line, the greater is the
magnitude r

• Coefficient ranges from -1 ≤ r ≤ 1


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

Assumption

• Observations are independent

• Relationship b/w two variables are linear

• Both variables should be normal distributed


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

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


Scatter diagram

• Each pair of variables is represented in scatter
diagram by a dot located at the point (x,y)


Scatter diagram - Merits

• Simple method

• Easy to understand

• Uninfluenced

• First step


Scatter diagram - Demerits

• Does not establish exact degree of correlation

• Qualitative method

• Not suitable for large sample


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

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

Assumption

• Observation are independent

• Samples are randomly selected

• The measurement scale is at least ordinal


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


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

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

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


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

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)


Multiple regression

• The dependent variable is continuous and
follows normal distribution

• Independent variable can be quantitative as
well as qualitative


9.testing of hypothesis

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