Statistics tests and Probablity
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The document discusses various statistical tests used to analyze quantitative and qualitative data, including the t-test, chi-square test, and tests of significance. It explains that the t-test is used to compare the means of two groups when the sample size is less than 30, while the z-test is used when the sample size is 30 or more. The chi-square test can be used to determine if there is an association between two variables and to compare proportions between groups. Key aspects like type I and type II errors in hypothesis testing and interpreting p-values are also summarized.
Statistics tests and Probablity
- 1. By Dr Aijaz Ahmed Sohag MSc (Env:Sc),M.A.S(H.S.A.),MBA(Health Mgt),MPH , PhD Prep by: Abdul Wasay Baloch Amna Inayat Medical College Statistic Test
- 2. Test of Significance ātā Test ļSummary points about ātā Test ļGosset in Dublin (1908) discovered it and called it āStudentā test later as ātā Test ļTo compare the means b/w groups t test is used ļUsed if sample size is less than 30 ļIf sample size is 30 or more than Z test is used ļPaired t ; when there is only one sample group is used and we take the means before and after interventions ļUnpaired ātā test ; where two sample groups are used ļFormula of t or Z test is=X1-X2/SE, where X1=mean of group 1, and X2=mean of group 2
- 3. Indication of ātā Test ļQuantitative data ļTo compare two Means ļ Random samples ļNormal distribution ļSample size is less than 30 ļāSā unknown ļContinuous Data ļParametric test
- 4. Types of ātā Test ļOne sample ātā Test ļTwo sample ātā Test ļPaired Sample ātā test
- 5. Level of Significance ļAlpha value: ļIt gives probability of incorrectly rejecting the null hypothesis when it is actually true ļTraditional values are used as 0.05,0.01 + 0.001 ļWhen a test statistics falls in the area of critical region the result is referred to as SIGNIFICANT
- 6. Conventions for Interpreting P values P Value Interpretation P> 0.05 Result not significant P<0.05 Result significant P<0.01 Result highly significant P<0.001 Result is very highly significant
- 7. Testing Null Hypothesis ļIn testing null hypothesis we have two decisions ļIt is false ā consequently rejected ļIt is true ā we fail to reject it ļIf we do decisions as above ā we do a correct decision ļIf we commit a mistake to decide incorrectly ā then we make/commit errors called alpha error and beta error
- 8. Errors of Hypothesis Testing ļTwo types of Errors 1. Type 1 or Ī± error : when we decide the null hypothesis is false, when it is actually true (no difference between two variables) ļ Rejecting the null hypothesis when it is true ļ Type 1 or alpha error(it is dangerous error) 1. Type 2 or Ī² error: when we decide that null hypothesis is true when it is actually false ļ Not rejecting the null hypothesis when it is actually false
- 9. 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
- 10. Chi Square Test ļAdvantages/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 ātā is not applicable)
- 11. Basis of Chi Square Test X2 ļMost tests involving quantitative data depend on x2 ļIn x2 we can test significance for many groups at the same time ļIn x2 actual no are used ļThe steps are ļStating null hypothesis ļCalculating x2 value ļFinding degree of freedom ļLooking for P value from x2 table ļAcceptability or rejecting the hypothesis
- 12. Chi Square Tests Test of significance of the difference b/w two proportion
- 13. 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 ā¢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
- 14. ļFirstly we assume or test the hypothesis in following ways ļConsidering the Null Hypothesis , that there was no differences b/w the effect of the two vaccines
- 15. Test the Null Hypothesis ļ proportion of people attacked will be 36/176=0.204 ļ Proportion of people not attacked will be 140/176=0.795 ļ From these proportion we calculate the expected no. of people attacked or cases by vaccine A 90*0.024=18.36 ļ Expected not attacked by vaccine A 90*0.795=71.55 ļ Similarly expected no. of attacked by Vaccine B 86* 0.204=17.544 ļ Expected no of non attacke by vaccine B 86*0.795=68.37
- 16. The expose (E) and Observed (O) are Vaccine A Attacked Non Attacked O=22 E=18.36 O-E=22-18.36= 3.64 O=68 E=71.55 O-E = 68-71.55= -3.55 Vaccine B Attacked Non Attacked O=14 E=17.54 O-E=14-17.54= -3.54 O=72 E=68.37 O-E= 72-68.37= 3.63
- 17. ļBy applying the Chi Square test ļX2 = ā (O-E)/ E ļCombining 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 ļB)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) =1 ļc) Referring Probability Table ļ by referring Chi Square Probable table having d.f 1 against probability of 0.05 =3.84
- 18. ļSince 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 ļThis test is valid only if the expected no.of each cell is not less than 2
- 19. Pakistan Demographic & Health Survey 06-07 ļ About apprehension/non fulfillment of MDG on improved maternal health ļ 96% women know about contraceptive knowledge, 22% using that ļ One in four unmarried women has unmet need for family planning ļ Most widely used method is Male Sterilization ļ In Sindh women of age 15-49 is 27% ļ Drop in total fertility rate from 5.4 children born to mother 90-91 to 4.1 children in 2006-07 ļ 1/3rd birth taken place within 24 months of previous birth which can be cited for increased Child Mortality ļ More than 9 of every 100 children die before 5th birthday ļ IMR in Sindh 8.1% while 7.8% in rest of country ļ MMR in Sind Ā¾ deaths in every 100,000. 20% of female deaths due to Maternal causes
- 20. ļHalf of birth by DAI, 39% by skilled doctor, nurse, midwife, LHV ļ47% children between 12 and 23 months receive all vaccines ļ02%of children under 5 ļ02% pregnant women sleep under net ļThe global population is growing by 80 miion people per year, 90% of it in poorer countries ļIn 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 ļAgriculture accounts for 70% of the withdrawals, a figure that reaches more than 90% in some developing countries
- 21. PROBABILITY ļThe probability of an event is denoted by P ļProbabilities are usually expressed as decimal fractions, not as percentages and must lie b/w zero (zero probability) and one (absolute probability) ļIf 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) ļ5 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
- 22. Example : FREQUENCY RATE OF DIABETES WAS DEFINITELY HIGHER AMONG OBESE ļBy calculating P-value, the statistical association between exposure status and occurrence of diabetes is ascertained ļTest of significance will depend upon the variables under investigation ļIf 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. ļHowever, 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- test ļIf p- value is less than or equal to equal to 0.05, it is regarded as āstatistically significantā
- 23. ļSmaller the p- value , the greater the statistical significance ļThe smaller the P value, the greater the statistical significance or probability that the association is not due to chance alone. ļHowever, statistical association (P value) does not imply causation. ļ ļP= 0.05 ( just significant at 5 percent level) ļP<0.05 (significant at 5 percent level) ļP>0.05 (not significant at 5 percent level)