Introduction to Inferential Statistics. Jaranit Kaewkungwal, Ph.D. Faculty of Tropical Medicine Mahidol University

Transcription

1 Introduction to Inferential Statistics Jaranit Kaewkungwal, Ph.D. Faculty of Tropical Medicine Mahidol University 1

2 Data & Variables 2

3 Types of Data QUALITATIVE Data expressed by type Data that has been described QUANTITATIVE Data classified by numeric value Data that has been measured or counted QUALITITATIVE and QUANTITATIVE data are not mutually exclusive Adapted from: Dr. Craig Jackson, University of Central England

4 Types of Data: Qualitative (Categorical) Data NOMINAL DATA values that the data may have do not have specific order values act as labels with no real meaning Binomial: two possible values (categories, states) Multinomial: more than two possible values (categories, states) e.g. Health status healthy =1 sick=2 e.g. Treatment new regimen = 1 standard regimen = 2 e.g. hair colour brown =1 blond =2 black =100 ORDINAL DATA values with some kind of ordering data that has been measured or counted e.g. social class: upper=1 middle = 2 working = 3 e.g. glioblastoma tumor grade: e.g. position in a race: Adapted from: Dr. Craig Jackson, University of Central England

5 Types of Data: Quantitative Data DISCRETE distinct or separate parts, with no finite detail e.g children in family CONTINUOUS between any two values, there would be a third e.g between meters there are centimetres INTERVAL equal intervals between values and an arbitrary zero on the scale e.g temperature gradient RATIO equal intervals between values and an absolute zero e.g body mass index Adapted from: Dr. Craig Jackson, University of Central England

6 Temperature Levels of Variables White Hot Dangerous 80 o C Unpleasant 60 o C Unsafe Red Hot Uncomfortable 40 o C Tolerable Comfortable 20 o C Safe Cold Cold 10 o C Adapted from: Dr. Craig Jackson, University of Central England

7 Examples of Data Coding Nominal/Cat. Var Exclude from Analysis? Ordinal/Cat. Var 7

8 Examples of Data Coding Cont. Cont

9 Example of Descriptive Statistics 9

10 Constant vs. Variable Variables are the specific properties that have the ability to take different values. Constants are the specific properties that cannot vary or won t be made to vary. 10

11 Terminology - Variables INDEPENDENT (syn: treatment, experimental, predictor, input, exposure, explanatory variable) is a stimulus or activity that is identified or manipulated to predict the dependent variable; they are considered as the causal factors, or that you may manipulate. e.g. new drug, working hours, exposure, worker attitudes, policies ies DEPENDENT (syn: Effect, criterion, criterion measure, outcome, output variable) is a response that the researcher wanted to predict; they are considered as the outcomes of the treatments or the responses to changes in the independent variables. e.g. Symptomotology,, productivity, accident rates, attitudes, health status, performance on neuropsychological test Adapted from: Dr. Craig Jackson, University of Central England

12 Terminology - Variables CONTROLLED Extraneous variable is a variable that has a potential to distorts the relationship between dependent and independent variables. Controlled extraneous variables are recognized before the study is initiated and are controlled in the design and selection criteria. Uncontrolled extraneous variables are recognized before the study is initiated or, sometimes, even if recognized cannot be controlled in the design and selection phase. Usually an attempt is made to assess and adjust them through sophisticated statistical tools. e.g., Working hours, temperatures, extraneous exposure, diet, class, income, Ambient noise and temperature in testing room Adapted from: Dr. Craig Jackson, University of Central England

13 Study Variables Independent Variables & Dependent variables X Y X (independent) Y (dependent) Extraneous variable X X (independent) Y (dependent) X2 (independent) 13

14 Study Variables Confounding Variable - When the effects of two or more variables cannot be separated. 14

15 Study Variables Confounding Variable - When the effects of two or more variables cannot be separated. BUT... Condom STD rate Yes 55/95 (61%) Use No 45/105 (43%) Condom Use increases the risk of STD STD rate # Partners < 5 Condom Yes 5/15 (33%) Use No 30/82 (37%) # Partners > 5 Condom Yes 50/80 (62%) Use No 15/23 (65%) Explanation: Individuals with more partners are more likely to use condoms. But individuals with more partners are also more likely to get STD. 15

16 Example of Study Variables Dependent Var: Infant//Child Growth Indepependent Var: Adult Fatness Extraneous Var: - Confouding Var (Adj,/Controlled Var) : Child Age, Adult Age, Socio-economic, Smoking, Physical Acitivity, etc. - Uncontrolled Var: Sex (male & female) 16

17 Bias & Chance 17

18 Measuring Outcomes: Observed vs. Truth Possible Explanations of Outcome Measured Bias Chance Truth Observed = Truth + Error Systematic error + Random error (Bias) (Chance) 18

19 Bias vs. Chance 19

20 Bias vs.chance Bias: A process at any stage of inference tending to produce results that depart systematically from the true values. Chance: The divergence of an observation on a sample from the true population value in either direction. The divergence due to chance alone is called random variation Bias and chance- are not mutually exclusive. 20

21 Bias vs.chance A well designed, carefully executed study usually gives results that are obvious without a formal analysis and if there are substantial flaws in design or execution a formal analysis will not help. Johnson AF. Beneath the technological fix. J Chron Dis 1985 (38),

22 Free kick Chance 2.5 % 50% 50% 68% 95% 2.5 % 22 Probability of being hit

23 Free kick Chance 2.5 % 50% 50% 68% 95% 2.5 % 23 Probability of getting goal

24 Normal Distribution in Descriptive Statistics Standard Score Raw Score X = 30; SD = 5 24

25 Types of Statistical Methods 25

26 Types of Statistics By Level of Generalization Descriptive Statistics Inferential Statistics Parameter Estimation Hypothesis Testing Comparison Association Multivariable data analysis Sampling Techniques Generalization/ Inferential Statistics By Level of Underlying Distribution Parametric Statistics Non-parametric Statistics 26

27 Descriptive Statistics 27

28 Descriptive Statistics Measure of Location (Categorical Vars) Frequency ( f ) Female 240 Male Count 210 Female Male Gender Measure of Location (Continuous Vars) Mean Average Median Mid-point Mode x = i = The Most Frequent n x i 1 or μ = n x i = 1 N X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 n i X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 ( ) 28

29 Descriptive Statistics Measure of Spread Range Max - Min Standard Deviation / Variance X i s deviate from Mean S or = n ( x i X ) i = 1 n - 1 σ = n ( x i μ ) i = 1 N x 29

30 Descriptive Statistics Measure of Shape Normal Distribution Skewed Distribution Positively skewed Negatively skewed 30

31 Inferential Statistics 31

32 Inferential Statistics Purpose of Inferential Statistics Generalisabiliy of research results from Sample Statistics to Population Parameters Types of Inferential Statistical Methods Parameter Estimation - to estimate the range of values that is likely to include the true value in population X μ proportion Π Hypothesis Testing - to ask whether an effect (difference) is present or not among different groups Ho: X 1 = X 2 Ho: μ 1 = μ 2 Ho: r xy = 0 Ho: ρ xy = 0 32

33 Sampling Distribution =26 =30 =25 SD = 9.1 SD = 11.3 SD = 12.2 μ = Inferential Statistics Inferential Statistics Sampling Method Sampling Method x 1 x 2 x 3

34 Sampling Distribution (Normal Distribution) in Parameter (μ)( Estimation Confidence Limits of μ : X ± Z α/2,ν S X Standard error 95% CI of μ : X ± (1.96 * (SD/ n)) 25 ± (1.96 * (12.2/ 100)) μ = 25 (22.6 to 27.4, 95% CI) n = 100 =26 =30 =25 SD = 9.1 SD = 11.3 SD = 12.2 μ = 27 N = 10,000 34

35 Parameter Estimation 35

36 Point Estimates: Single values (Mean, Variance, Correlation, treatment effect, relative risk, etc.) representing characteristics in the whole population Parameter Estimation Interval Estimates: Ranges of values, usually centered around point estimates, indicating bounds within which we expect the true values for the whole population to lie (stability of the estimate) SD = 0.75 Average: Example Confidence Limits of μ : X ± Z S α / 2, υ X 36

37 Parameter Estimation Point estimates & Confidence Intervals Point estimates and confidence intervals are used to characterise the statistical precision of any rate (incidence, prevalence), comparisons of rates (e.g., relative risk), and other statistics. US adults have used unconventional therapy = 34% (31% - 37%, 95%CI) Sensitivity of clinical examination of splenomegaly = 27% (19-36%, 95%CI) Relative risk of lung cancer of smoker vs. non-smoker = 5.6 ( , 95%CI) Relative risk of HIV infected of male vs. female = 2.1 ( , 95%CI) 37

38 Parameter Estimation (Normal Distribution) Confidence Limits of μ : X ± Z α/2,ν S X Standard error 95% CI of μ : X ± (1.96 * (SD/ n)) 25 ± (1.96 * (12.2/ 100)) μ = 25 (22.6 to 27.4, 95% CI) n = 100 =26 =30 =25 SD = 9.1 SD = 11.3 SD = 12.2 μ = 27 N = 10,000 38

39 Parameter Estimation Consideration in confidence level of the estimate Select a cut-point for CI Confidence Interval, usually set at 95% CI, can be interpreted such that - if the study is unbiased and repeated 100 times, there is 95% chance that the true value is included in these intervals of the 100 samples 95% CI (from Sample 1) True Value (in population) 95% CI (from Sample 2) 95% CI (from Sample 3) 39

40 Hypothesis Testing - Comparisons 40

41 Hypothesis Testing Hypothesis & Tail of the test One-sided vs. Two-sided Test Two-sided test: Ex Ho: Outcome 1 = Outcome 2 Ha: Outcome 1 Outcome 2 One-sided test: Ex Ho: Outcome 1 Outcome 2 Ha: Outcome 1 > Outcome 2 Ho: Outcome 1 Outcome 2 Ha: Outcome 1 < Outcome 2 O1<O2 O1=O2 O1>O2 2.5% 95% 2.5% O1<O2 O1 >= O2 5% 95% 41

42 Hypothesis Testing Basic steps in hypothesis testing 42

43 Hypothesis Testing Ho: μ 1 = μ 2 Ho: μ 1 μ 2 = 0 Ha: μ 1 μ 2 = 0 Not Reject Ho!! μ 1 = μ 2 μ 1 μ 2 43

44 Hypothesis Testing Ho: μ 1 μ 2 = 0 Ha: μ 1 μ 2 = 0 Reject Ho!! μ 1 < μ 2 μ 1 μ 2 44

45 Hypothesis Testing H 0 : μ 1 μ 2 = 0 H a : μ 1 μ 2 = 0 given n = very large at α = 0.05 Reject H 0!! μ 1 > μ 2 at α = 0.01 Not Reject H 0!! μ 1 = μ 2 α / 2 = p-value = 0.04 α / 2 =

46 Type I & Type II errors H 0 : G 1 = G 2 Hypothesis Testing Reality/Truth H 0 True (G 1 =G 2 ) H 0 False (G 1 <>G 2 ) Decision Accept H 0 (G1=G2) Reject H 0 (G1<>G2) Correct Type II Error Confidence : 1 - α A B 0.99, , 0.20 Type I Error α C D β Correct Power : 1 - β 0.01, ,

47 Choosing the Right Statistical Procedure 47

48 Choosing the Right Data Analysis Procedure Basic questions that you should have answers before you can choose the correct test are the following: What is the purpose of the analysis? - describe the data; Or - compare groups of data to make decisions; or - examine the association between variables for prediction or forecasting? Is the distribution of the data approximately normal? Is the sample size large enough that the Central Limit Theorem will allow a normality assumption? 48

49 Selecting a Statistical Method Goal Type of Outcome Data of the Continuous Categorical Binomial Survival analysis (from Gaussian Continuous Time Population) (Non-Gaussian) Describe Value of Data (1 Group) Mean, SD Median, Interquartile range Proportion (Percent) Kaplan- Meier survival curve Compare Value of Data vs. Hypothetical Value Onesample t- test Wilcoxon s test Chi-square (χ 2 ) or Binomial/ Runs test (1 Group) 49

50 Selecting a Statistical Method Goal Type of Outcome Data of the Continuous Categorical Binomial Survival analysis (from Gaussian Continuous Time Population) (Non-Gaussian) Compare Values 2 Grps of Indept >2 Grps Grps. Unpaired t- test One-way ANOVA Mann- Whitney test Kruskal- Wallis test χ 2 test, Fisher s Exact, χ 2 test Log-rank / Mantel- Haenszel Cox Prop Haz.Reg. Compare Values 2 Grps of Paired >2 Grps Grps/Vars. Paired t-test Repeated measures ANOVA Wilcoxon s test Friedman s test McNemar s χ 2 test Cochrane s Q test Condt n Prop Haz.Reg. Condt n Prop Haz.Reg. 50

51 Selecting a Statistical Method Goal Type of Outcome Data of the Continuous Categorical Binomial Survival analysis (from Gaussian Continuous Time Population) (Non-Gaussian) Quantify Association Values of Two variables Pearson s Correlation Spearman s Correlation Contingency coefficient, Crude Odds Ratio, Relat v Risk Predict Value of Outcome Var: from 1 Var (Simple Reg. ) from > 2 Vars (Multiple Reg.) Linear or Non-linear Regression. Nonparametric Regression Logistic Regression Cox s Proportional Haz. Reg. 51

52 Selecting a Statistical Method Goal Type of Outcome Data of the Continuous Categorical Binomial Survival analysis (from Gaussian Continuous Time Population) (Non-Gaussian) Measures of Agreement Values from Two Raters/Methods Pearson s Correlation Weighted Kappa (κ) Weighted Kappa (κ) Agreement rate Cohen s κ ICC Measures of Validity Values from Two Raters/Methods ANOVA Factor Analysis Nonparametric ANOVA Factor Analysis χ 2 Sensitivity Specificity ROC curve 52

53 Example of Inferential Statistical Methods 53

54 Example of Parameter Estimates & 95%CI 54

55 Example of Parameter Estimates & 95%CI 55

56 Example of Comparisons 56

57 Example of Comparisons 57

58 58

59 Example of Comparisons 59

60 Example of Comparisons 60

61 Example of Association 61

62 Example of Association 62

63 Example of Association 63

64 Example of Association 64

65 Example of Association 65

66 Example of Association 66

67 Dose-Response 67

68 68

69 Example of Parameter Estimates 69

70 Example of Parameter Estimates Figure 2. Survival from time of human immunodeficiency virus (HIV) infection of 194 CSWs. A, Overall. B, By serum virus load (HIV type 1 RNA copies/ml). Each curve is truncated when <10 women remain in that group <10, , ,000 Survival (%) A Survival (%) >100,000 B Months Months 70

71 Example of Association Table 2. Survival from time of infection of 194 HIV-infected CSWs Charactersitcs No. of No. of Patients 7-Year survival, Rate ratio patients who died % % (95% CI)* (95% CI)** Age at Infection, years <= (29.5) 72.3 ( ) Referent >= (39.3) 63.3 ( ) 1.50 ( ) Sex work Brothel (34.0) 69.6 ( ) 1.34 ( ) Nonbrothel (34.3) 62.9 ( ) Referent Oral contraceptive use Yes (32.1) 69.6 ( ) 0.83 ( ) No (36.6) 67.1 ( ) Referent Depot medroxyprogesterone use Yes (32.7) 70.9 ( ) 0.78 ( ) No (34.5) 67.8 ( ) Referent Infection status Seroconverted 34 7 (20.6) *** 1.42 ( ) Seropositive at enrl (36.9) 69.6 ( ) Referent Viral load, HIV-1 RNA copies/ml. >1000, (70.6) 34.5 ( ) ( ) 10, , (33.6) 70.3 ( ) 4.63 ( ) <10, (8.5) 92.5 ( ) Referent Total (34.0) 68.7 ( ) * Survival analysis ** Cox proportional hazard model *** Insufficient follow-up time to this m ore recent converted group; 5-year survival = 77.8 ( ) % 71

72 Statistical Methods: Multi-variable Data Analysis 72

73 Multi-variable Data Analysis Causes/Exposures vs. Outcomes Example: Causes of Tuberculosis Malnutrition Vaccination crowding Genetic Susceptible Host Exposure to Mycobacterium Infection Tissue Invasion and Reaction Tuberculosis Risk factors for tuberculosis (Distant from Outcome) Mechanism of Tuberculosis (Proximal to Outcome) 73

74 Multi-variable Data Analysis Causes/Exposures vs. Outcomes Example: Relationship between risk factors and disease : hypertension ( BP) and congestive heart failure (CHF). Hypertension causes many diseases, including congestive heart failure, and congestive heart failure has many causes, including hypertension. 74

75 Example of MDA Risk (never) = 7/47 =.149 Risk (sporadic) = 9/35 =.257 RR (sporadic vs.never) = 25.7/14.9 =

76 Example of MDA 76

77 Example of MDA Table 3. Mortality from time of first CD4 T lymphocyte count of 157 HIV-infected CSWs (125 women were HIV seropositive at study enrollment and 32 seroconverted during study) Charactersitcs No. of No. of Pts 5-Year survival, Rate ratio Adjusted rate ratio patients who died % % (95% CI)* (95% CI)** (95% CI)*** Initial CD4 lymphocyte, cells/μl < (93.3) ( ) 15.5 ( ) (38.6) 63.4 ( ) 2.46 ( ) 1.42 ( ) > (18.5) 84.7 ( ) Referent Referent Viral load, HIV-1 RNA copies/ml. >1000, (76.7) 26.7 ( ) 13.9 ( ) 12.5 ( ) 10, , (34.8) 65.0 ( ) 3.87 ( ) 3.42 ( ) <10, (10.5) 96.7 ( ) Referent Referent Total (36.9) 64.6 ( ) * Survival analysis ** Cox proportional hazard model *** Cox proporational hazard model adjusted for initial CD4 lymphocyte count and virus load 77

78 The End of Inferential Statistics 78

79 Bias vs.chance: Observed & Truth in Statistical Methods Observed = Truth + Error Example of Statistical Analysis: Chi-Square test: χ 2 =Σ(Ο Ε) 2 /Ε Kappa Statistics: Observed agreement - Expected Agreement κ = Obs Agreemt - Expct Agreemt 1 - Expected Agreement Systematic error + Random error OBSERVED exposed EXPECTED exposed Yes No Yes No Outcome Yes Yes No Outcome No

80 Bias vs.chance: Observed & Truth in Statistical Methods Observed = Truth + Error Systematic error + Random error Example of Statistical Analysis: 190 ε Analysis of variance: systolic μ τ ε τ F = σ 2 (τ+ε) /σ2 ε <=20 3 agegrp >= 41 age group

81 Bias vs.chance: Observed & Truth in Statistical Methods Observed = Truth + Error Example of Statistical Analysis: Regression Analysis: Y = Y + ε Y = α + β 1 X 1 Systematic error + Random error CHD Mortality per 10, ε Y=13, X =6 (Y = 11, X = 6) Cigarette Consumption per Adult per Day 81