【医学】《医学统计学》6-Chisquare-test.ppt

Medical Statistics(full English class)Ji-Qian FangSchool of Public Health Sun Yat-Sen UniversityChapter 10 Statistical Analysis of Enumeration Data10.1 Statistical Description for enumeration data Absolute measure:Absolute measure:l The numbers counted for each The numbers counted for each category(frequencies)category(frequencies)l The absolute measure can hardly be The absolute measure can hardly be used for comparison between used for comparison between different populations.different populations.1.Relative measure Three kinds of relative measures:Three kinds of relative measures:Frequency(Proportion)Frequency(Proportion)Intensity(Rate)Intensity(Rate)Ratio Ratio(1)Relative FrequencyNote:The Chinese text book is wrong!It is not“rate”!It is proportion or frequency!Example 10-1(P.304,revised)Question:Which grade has the most serious condition of myopias?Prevalence rates describe:P(Myopia|First grade)P(Myopia|Second grade)P(Myopia|Third grade)Constitute among myopias describe:P(First grade|Myopia)P(Second grade|Myopia)P(Third grade|Myopia)Which grade has the most serious condition of myopias?Answer:P(Myopia|Third grade)=Maximum-The third grade has the highest prevalence of myopias P(Second grade|Myopia)=Maximum-Among the myopias,the absolute number of Second grade students is the highest.(2)IntensityExample A smoking population had followed up for Example A smoking population had followed up for Example A smoking population had followed up for Example A smoking population had followed up for 562833 person-years,346 lung cancer cases were 562833 person-years,346 lung cancer cases were 562833 person-years,346 lung cancer cases were 562833 person-years,346 lung cancer cases were found.found.found.found.The incidence rate of lung cancer in the smoking The incidence rate of lung cancer in the smoking The incidence rate of lung cancer in the smoking The incidence rate of lung cancer in the smoking population is:population is:population is:population is:Incidence rate=346/562833 =61.47 per 100,000 person-year Example The mortality rate of liver cancer in Guangzhou is 32 per 100,000 per year.In general,Denominator:Sum of the person-years observed in the periodNumerator:Total number of the event appearing in the periodUnit:person/person year,or 1/YearNature:the relative frequency per unit of time.(3)Ratio Ratio is a number divided by another related numberExamples Sex ratio of students in this class:No.of males:No.of females=52%Coefficient of variation:CV=SD/mean Ratio of time spent per clinic visit:Large hospital:Community health station =81.9 min.:18.6 min.=4.402.Caution in use of relative measuresa.a.The denominator should be big enough!Otherwise The denominator should be big enough!Otherwise the absolute measure should be used.the absolute measure should be used.Example:Out of 5 cases,3 were cured Example:Out of 5 cases,3 were cured 60%?60%?b.b.Attention to the population where the relative Attention to the population where the relative measure comes from.measure comes from.Mistake in the textbook(P.305):Mistake in the textbook(P.305):“Distinguish between constitutes and proportionDistinguish between constitutes and proportion”!?!?We should sayWe should say “Distinguish between Prevalence Distinguish between Prevalence rate and Constitute among patientsrate and Constitute among patients”Prevalence rate:Population is the students in thePrevalence rate:Population is the students in thesame grade same grade Constitutes:Population is all the patients Constitutes:Population is all the patients The above two frequency distributions reflect two populations of all patients;To describe the prevalence rate,one has to look at the general population;c.Pooled estimate of the frequencyc.Pooled estimate of the frequency Pooled estimate Pooled estimate =numerators/numerators/denominators denominators Example:Example:The prevalence of myopia among 3 grades The prevalence of myopia among 3 grades (15.16+15.89+18.37)/3 (15.16+15.89+18.37)/3 The prevalence of myopia among 3 grades The prevalence of myopia among 3 grades =(67+68+56)/(442+428+305)=(67+68+56)/(442+428+305)=192/1175 =192/1175 =16.34 =16.34d.d.Comparability Comparability between between frequencies frequencies or or between between frequency frequency distributions distributions Notice Notice the the balance balance of of other conditionsother conditionse.If the distributions of other variables are different,to improve the comparability,“Standardization”is needed.f.To compare two samples,hypothesis test is needed.(See Chi square test)The following will emphasize the above two points:Standardization Hypothesis test 3.Standardization for crude frequency or crude intensityCrude incidence rate of city A=28.96;Crude incidence rate of city B=35.03 -Strange!?They are not comparable!-Because the constitute are quite different Table 10-3 Incidence rates of infectious diseases,children of two citiesStandardized incidence rate of city A=793/24767=32.02 Standardized incidence rate of city B=3523/24767=21.12 Two steps:lSelect a standard population taking as“weight”lWeighted average of the actual incidence ratesdirect standardization rateKnown:Age specific populations Ni1,Ni2;Total no.of deaths Di1=432,Di2=210 Select a set of standard mortality rates Standard mortality ratio:SMR1=Di1/Ni1Pi=432/100.67=4.2912(smoker)SMR2=Di2/Ni2Pi=210/100.67=0.8620(non-smoker)Standardized mortality rate P1=34.60 SMR1=148.48(1/105),P2=34.60 SMR2=29.83(1/105)10.2 Statistical Inferencefor Enumeration Data1.Sampling error of frequency Example Suppose the death rate is 0.2,if the ratsare fed with a kind of poison.What will happen when we do the experiment on n=1,2,3 or 4 rat(s)?In general,Supposed the population proportion is ,sample size=n l The frequency is a random variablelWhen is unknown and n is big enough,is approximately equal to Example 10-5 HBV Surface antigen.200 people were tested,7 positive.If the sample size n is big enough,and observed frequency is p,then we have approximately 2.Confidence Interval of Probability If the sample size n is big enough,and observed frequency is p,then95%Confidence interval 99%Confidence interval Example 10-5 HBV Surface antigen.200 people were tested,7 positive.3.The hypothesis testing of proportion (u test)1.Comparison of sample proportion and population proportion Example 10.6 Cerebral infarction Cases Cure rateNew Method 98 50%Routine 30%lStatistic ulDecision rule If ,then reject Otherwise,no reason to reject (accept )Since ,reject 2.Comparison of two sample proportionsExample 10.7 Carrier rate of Hepatitis B City:522people were tested,24 carriers,4.06%(population carrier rate:1)Countryside:478people were tested,33 carriers,6.90%(population carrier rate:2)lPooled estimatelStandard error of P1-P2 lStatistic ulDecision rule If ,then reject Otherwise,no reason to reject (accept )Since ,not reject SummarylThe parameter estimation and hypothesis testing of proportion are based on the normal approximation(when sample size is big enough)lHow big is enough?By experience,n 5 and n(1-)5l If the sample size is not big,u test cant be used and there is no t-test for proportion.(see more detailed text book)10.3 Chi-square test TheThe u u test test can only be used for can only be used for comparing comparing with a given with a given 0(one sample)(one sample)or comparing or comparing 1 with 2(two samples).(two samples).If we need to compare more than If we need to compare more thantwo samples,two samples,Chi-square testChi-square test is widelyis widelyused.used.1.Basic idea of 2 testlGiven a set of observed frequency distribution Given a set of observed frequency distribution A A1 1,A A2 2,A A3 3 to test whether the data follow certain theory.to test whether the data follow certain theory.lIf the theory is true,then we will have a set If the theory is true,then we will have a set of theoretical frequency distribution:of theoretical frequency distribution:T T1 1,T T2 2,T T3 3 lComparing Comparing A A1 1,A A2 2,A A3 3 and and T T1 1,T T2 2,T T3 3 If they are quite different,then the theory might If they are quite different,then the theory might not be true;not be true;Otherwise,the theory is acceptable.Otherwise,the theory is acceptable.Example10-8 Example10-8 AcuteAcute lower respiratory infectionlower respiratory infection TreatmentEffectNon-effectTotalEffect rateDrug A68(64.82)a6(9.18)b74(a+b)91.89%Drug B52(55.18)c11(7.82)d63(c+d)82.54%Total120(a+c)17(b+d)13753.59%(2)Chi-square test for 2 2 table H H H H0 0:1 1 1 1=2 2 2 2,H H H H1 1:1 1 1 1 2 2 2 2,=0.05 =0.05 =0.05 =0.05 n To calculate the theoretical frequencies To calculate the theoretical frequencies If If If If H H H H0 0 is true,is true,1 1 1 1=2 2 2 2 120/137120/137120/137120/137 T T T T11111111=74=74=74=74 120/137=64.82,120/137=64.82,120/137=64.82,120/137=64.82,T T T T21212121=63=63=63=63 120/137=55.18120/137=55.18120/137=55.18120/137=55.18 T T T T12121212=74=74=74=74 17/137=9.18,17/137=9.18,17/137=9.18,17/137=9.18,T T T T22222222=63=63=63=63 17/137=7.8217/137=7.8217/137=7.8217/137=7.82 lTo compare A and T by a statistic To compare A and T by a statistic 2 2 If If H H0 0 is true,is true,2 2 follows a chi-square follows a chi-square distribution.distribution.=(row-1)(column-1)=(row-1)(column-1)If the If the 2 2 value is big enough,we doubt value is big enough,we doubt about about H H0 0,then reject,then reject H H0 0!To Example10-8To Example10-8 ,=(row-1)(column-1)=(2-1)(2-1)=1,=(row-1)(column-1)=(2-1)(2-1)=1,2 20.05(1)0.05(1)=3.84,=3.84,Now,Now,2 2=2.7343.84,=2.7340.05,0.05,H H0 0 is not rejected.is not rejected.We have no reason to say the effects of We have no reason to say the effects of two treatments are different.two treatments are different.For For 2 2 2 2 table table,there is a specific,there is a specific formula of chi-square calculation:formula of chi-square calculation:To Example10-8To Example10-8 ,Large sample is requiredLarge sample is required (1)(1)N N 40,40,T Ti i 5,5,N N 4040(2)If(2)If n n 40 or 40 or T Ti i 1,1,2 2 test is not applicable test is not applicable(3)If(3)If N N 40,40,1 1 T Ti i 5,needs adjustment:40)For large sample(b+c40)Otherwise,needs adjustmentOtherwise,needs adjustmentl If If the the 2 2 value is value is too big,then reject too big,then reject H H0 0 Example10-10Example10-10:=1,=1,4.924.923.84,3.84,P P0.05,0.05,H H0 0 is rejected is rejectedConclusion:There is significant difference in Conclusion:There is significant difference in positive rates between the two diagnosis positive rates between the two diagnosis methods.methods.Since Since P PA A 2,2,“H H0 0 is rejected is rejected”only means there only means there is difference among some groups.Does not is difference among some groups.Does not necessary mean that all the groups are different.necessary mean that all the groups are different.(3)(3)The 2 test requires large sample:By experience,lThe theoretical frequencies should be greater than 5 in more than 4/5 cells;lThe theoretical frequency in any cell should be greater than 1.Otherwise,we can not use chi-square test directly.Otherwise,we can not use chi-square test directly.If the above requirements are violated,what should we do?(1)Increase the sample size.(2)Re-organize the categories,Pool some categories,or Cancel some categories Think:In fact,it is not appropriate to use a Chi-square test for Example 10-10 in the textbook.Why?更多精品文档，请点击：更多精品文档，请点击：http:/ 人的一生，总是难免有浮沉。不会永远如旭日东升，也不会永远痛苦潦倒。反复地一浮一沉，对于一个人来说，正是磨练。因此，浮在上面的，的，不必骄傲；沉在底下的，更用不着悲观。必须以率直、谦虚的态度，乐观进取、向前迈进。松下幸之助 成功人生八大心态成功人生八大心态一、成功者的心态（老板的心态）一、成功者的心态（老板的心态）天生赢家天生赢家为成功而生为成功而生责任责任老板的心态老板的心态二、学习的心态二、学习的心态学习是通向未来的护照学习是通向未来的护照空杯归零空杯归零低头低头以身作则以身作则盗版盗版学到，做到，教到学到，做到，教到三、合作的心态三、合作的心态环境改变人环境改变人1+1 21+1 2雁群飞翔雁群飞翔-合作合作天时天时-地利地利-人和人和四、付出的心态四、付出的心态五、积极五、积极 的心态的心态平常心平常心平等心平等心包容心包容心六、吃苦耐劳的心态六、吃苦耐劳的心态七、坚持的心态七、坚持的心态八、感恩的心态八、感恩的心态感恩国家、社会、董事长、推荐人感恩国家、社会、董事长、推荐人伙伴伙伴父母父母另一半另一半子女子女