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 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)


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


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 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


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

• 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


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 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


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 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


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 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


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 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


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


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 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


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 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


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


Education

9.testing of hypothesis