Statistics tests and Probablity

By Dr Aijaz Ahmed Sohag MSc (Env:Sc),M.A.S(H.S.A.),MBA(Health

Mgt),MPH , PhD

Prep by: Abdul Wasay BalochAmna Inayat Medical College

Statistic Test

Test of Significance ‘t’ TestSummary points about ‘t’ Test

Gosset in Dublin (1908) discovered it and called it ‘Student’ test later as ‘t’ Test

To compare the means b/w groups t test is usedUsed if sample size is less than 30 If sample size is 30 or more than Z test is usedPaired t ; when there is only one sample group is used

and we take the means before and after interventionsUnpaired ‘t’ test ; where two sample groups are usedFormula of t or Z test is=X1-X2/SE, where X1=mean of

group 1, and X2=mean of group 2

Indication of ‘t’ TestQuantitative dataTo compare two Means Random samplesNormal distributionSample size is less than 30“S” unknownContinuous DataParametric test

Types of ‘t’ TestOne sample ‘t’ TestTwo sample ‘t’ TestPaired Sample ‘t’ test

Level of SignificanceAlpha value:

It gives probability of incorrectly rejecting the null hypothesis when it is actually true

Traditional values are used as 0.05,0.01 + 0.001When a test statistics falls in the area of critical

region the result is referred to as SIGNIFICANT

Conventions for Interpreting P values

P Value InterpretationP> 0.05 Result not significant

P<0.05 Result significant

P<0.01 Result highly significant

P<0.001 Result is very highly significant

Testing Null HypothesisIn testing null hypothesis we have two decisionsIt is false – consequently rejectedIt is true – we fail to reject itIf we do decisions as above – we do a correct

decisionIf we commit a mistake to decide incorrectly – then

we make/commit errors called alpha error and beta error

Errors of Hypothesis Testing Two types of Errors1.Type 1 or α error : when we decide the null

hypothesis is false, when it is actually true (no difference between two variables) Rejecting the null hypothesis when it is true Type 1 or alpha error(it is dangerous error)

2.Type 2 or β error: when we decide that null hypothesis is true when it is actually false Not rejecting the null hypothesis when it is actually

false

inference Accept it Reject it

Null Hypothesis Correct decision Type 1 error

NH is false Type 2 Error Correct Decision

In Medical studies Type 1 error is more serious as

compared to Type 2 errors

Chi Square TestAdvantages/Rationale

a)Alternate method: to testify significance of difference between two proportions

b)To determine whether there is some association between two variables

c)It is applicable to qualitative data (where ‘t’ is not applicable)

Basis of Chi Square Test X2Most tests involving quantitative data depend on x2In x2 we can test significance for many groups at

the same timeIn x2 actual no are usedThe steps are

Stating null hypothesisCalculating x2 valueFinding degree of freedomLooking for P value from x2 tableAcceptability or rejecting the hypothesis

Chi Square Tests

Test of significance of the difference b/w two proportion

An Example:Trial of 2 whooping cough vaccines

Vaccine No of vaccinated

No of cases Non Attacked Total

A 2400 22 68 90

B 2300 14 72 86

Total 4700 36 140 176

•Apparently vaccine was B was superior to Vaccine A, to know whether the vaccine was really superior to vaccine A OR whether the diff was merely due to chance

Firstly we assume or test the hypothesis in following waysConsidering the Null Hypothesis , that there was no

differences b/w the effect of the two vaccines

Test the Null Hypothesis proportion of people attacked will be 36/176=0.204 Proportion of people not attacked will be

140/176=0.795 From these proportion we calculate the expected no.

of people attacked or cases by vaccine A 90*0.024=18.36 Expected not attacked by vaccine A 90*0.795=71.55 Similarly expected no. of attacked by Vaccine B 86*

0.204=17.544 Expected no of non attacke by vaccine B

86*0.795=68.37

The expose (E) and Observed (O) areVaccine A Attacked Non Attacked

O=22E=18.36

O-E=22-18.36= 3.64

O=68E=71.55

O-E = 68-71.55= -3.55

Vaccine B Attacked Non Attacked

O=14E=17.54

O-E=14-17.54= -3.54

O=72E=68.37

O-E= 72-68.37= 3.63

By applying the Chi Square testX2 = ∑ (O-E)/ ECombining O-E attacked case & non attacked case

= 3.64^2/18.36 + 3.55^2/71.55 + 3.54^2 / 17.54 + 3.36^2/68.37=0.72+0.17+0.71+0.19+1.7

B)Finding the degree of Freedom (d.f)- depends on no. of columns and rows in a table

d.f=(c-1) (r-1) = (2-1) (2-1)

=1c) Referring Probability Table

by referring Chi Square Probable table having d.f 1 against probability of 0.05

=3.84

Since the observed value in Chi Square table is much lower so the null hypothesis is true, hence Vaccine B is not superior to Vaccine A

This test is valid only if the expected no.of each cell is not less than 2

Pakistan Demographic & Health Survey 06-07

About apprehension/non fulfillment of MDG on improved maternal health 96% women know about contraceptive knowledge, 22% using that One in four unmarried women has unmet need for family planning Most widely used method is Male Sterilization In Sindh women of age 15-49 is 27% Drop in total fertility rate from 5.4 children born to mother 90-91 to 4.1

children in 2006-07 1/3rd birth taken place within 24 months of previous birth which can be

cited for increased Child Mortality More than 9 of every 100 children die before 5th birthday IMR in Sindh 8.1% while 7.8% in rest of country MMR in Sind ¾ deaths in every 100,000. 20% of female deaths due to

Maternal causes

Half of birth by DAI, 39% by skilled doctor, nurse, midwife, LHV

47% children between 12 and 23 months receive all vaccines

02%of children under 502% pregnant women sleep under netThe global population is growing by 80 miion people per

year, 90% of it in poorer countriesIn past 50 years extraction from rivers, lakes and aqiofers

has tripled to help meet population growth and demand for water intensive food such as rice cotton, dairy and meat products

Agriculture accounts for 70% of the withdrawals, a figure that reaches more than 90% in some developing countries

PROBABILITYThe probability of an event is denoted by PProbabilities are usually expressed as decimal

fractions, not as percentages and must lie b/w zero (zero probability) and one (absolute probability)

If the event is sure to occur, than p-value is 1(absolute probability), for e.g. all men sure to die. So probability is P = 100/100= 1(standard)

5 chances in 100=5/100 OR 1/20 OR 0.05. We can also say I chance in 20 is taken as cut off value

Example : FREQUENCY RATE OF DIABETES WAS DEFINITELY HIGHER AMONG OBESEBy calculating P-value, the statistical association between exposure status and occurrence of diabetes is ascertainedTest of significance will depend upon the variables under investigationIf we are dealing with discrete variables(cannot be expressed in decimals), the results are usually presented as rates and proportion, THAN test of significance usually adopted is the STANDARD ERROR OF DIFFERENCE BETWEEN TWO PROPORTION or CHI-SQARE TEST.However, if we are dealing with continuous variables(can be expressed in decimals e.g., age, blood pressure), the data will have to be grouped and test of significance used will be STANDARD ERROR OF DDIFFERENCE BETWEEN TWO MEANS, or t- testIf p- value is less than or equal to equal to 0.05, it is regarded as “statistically significant”

Smaller the p- value , the greater the statistical significance

The smaller the P value, the greater the statistical significance or probability that the association is not due to chance alone.

However, statistical association (P value) does not imply causation.

 P= 0.05 ( just significant at 5 percent level)P<0.05 (significant at 5 percent level)P>0.05 (not significant at 5 percent level)