# Correlation and regression

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1 1 Correlation and regression Yongjua Laosiritaworn Introductory on Field Epidemiology 6 July 2015, Thailand

2 Data 2

3 Illustrative data (Doll, 1955) 3

4 Scatter plot 4

5 Doll,

6 6

7 Correlation coefficient, r 7

8 Interpretation of r 8

9 r by hand 9

10 r by hand 10

11 11 Coefficient of determination (r2)

12 Cautions 12 Outliers Non-linear relationship Confounding (correlation is not causation) randomness

13 Outliers 13

14 Non-linear relaitonship 14

15 Confounding 15

16 Randomness 16

17 Example of questions 17 Estimate association between outcome and covariates How cardiovascular disease (CVD) associate with smoking [and body mass index (BMI), age, etc.] Control of confounder(s) How cardiovascular disease (CVD) associate with smoking after adjust for BMI, age, etc. Risk prediction How to predict probability of getting CVD given information of BMI, age, etc.

18 Result from bivariate analysis 18 Outcome is CVD, exposure is smoker CVD n=100 No CVD n=100 Odds ratio (95% CI) Smoker (1.80, 6.03) Alcohol drinker (1.01, 3.10) Confounder?

19 19 Correlation direction and strength

20 Sequence of analysis 20 Descriptive analysis Know your dataset Bivariate analysis Identify associations Stratified analysis Identify confounders and effect modifiers Multivariable analysis Control for confounders and manage effect modifiers

21 Regression 21 Regression is the study of dependence between an outcome variable (y) - the dependent variable, and one or several covariates (x) - independent variables Objective of using regression Estimate association between outcome and covariates Control of confounder(s) Risk prediction

22 Result from bivariate analysis 22 Outcome is CVD, exposure is smoker CVD n=100 No CVD n=100 p - value Mean MBI (sd) 25.6(1.5) 21.6(1.6) < Mean age (sd) 63.0(7.2) 56.1(7.1) < Confounder?

23 Stratify analysis 23 Outcome and Exposure a c b d Crude OR Adjusted OR i.j?? Variable1 a 1 c 1 b 1 d 1 a i c i OR 1 a 2 c 2 b 2 d 2 OR 2 OR i b i d i Variable2 a 1 c 1 b 1 a 2 a 1 c 1 b 1 a 2 d 1 OR 2.1 c 2 b 2 d 2 OR 2.2 a j c j b j d j OR 2.j a 1 c 1 b 1 a 2 c 2 b 2 a j c j b j d j d 1 OR 1.1 c 2 b 2 d 2 OR 1.2 a j c j b j d j OR 1.j d 1 OR i.1 d 2 OR i.2 OR i.j Variable a 1 b 1 OR c 1 d a 2 b 2 OR c 2 ad b 1 OR ac j 1 bd 1.1 j 1 OR ca j d 1.j 2 b j 2 OR c 2 da b 1 OR a c j b 1 d j OR c j ad 2 1.j c j b a 2 OR 2 d b 1 OR c 2 1 d a j b a j OR c j d 1.j 2 b 2 OR c j 2 ad b 1 OR ac j 1 bd 1.1 j 1 OR ca j d 1.j 2 b j 2 OR c 2 da b 1 OR a c j b 1 d j OR c j ad 2 1.j c j b 2 OR a 2 d b 1 OR 2 c 1 da j b j OR a c j d 1.j 2 b 2 OR j c 2 ad b 1 OR ac j 1 bd 1.1 j 1 OR ca j d 1.j 2 b j 2 OR c 2 da b 1 OR a c j b 1 d j OR c j ad 2 1.j c j b 2 OR 2 d a j c j b j d j OR 1.j

24 Regression 24 It is a representation/modeling of the dependence between one or several variables Type of models Linear regression Logistic regression Cox regression Poisson regression Loglinear regression Choices of the model depend on objectives, study design and outcome variables

25 Review of linear equation 25

26 x and y relationship example1 26 x y y = x

27 x and y relationship example2 27 x y y = 2x

28 x and y relationship example3 28 x y y = 2x + 1

29 General form of linear equation 29 x is independent variable y is dependent variable a is a constant or y-intercept The value of y when x = 0 b is a slope y = a + bx Amount by which y changes when x changes by one unit

30 Exercise: interpretation of slope 30 y = x x increase 1 y. y = 2x x increase 1 y. y = 2x + 1 x increase 1 y. y = -x + 1 x increase 1 y. y = -2x - 1 x increase 1 y.

31 Linear regression 31

32 Linear regression 32 Given an independent variable (x) and a dependent continuous variable (y), we computed r as a measure of linear relationship We want to be able to use our knowledge of this relationship to explain our data, or to predict new data. This suggests linear regression (LR), a method for representing the dependent variable (outcome) as a linear function of the independent variables (covariates) Outcome is continuous variable Covariate can be either continuous or categorical variable

33 Linear regression 33 Simple linear regression (SLR) A single covariate Multiple linear regression (MLR) Two or more covariate (with or without interactions)

34 Mathematical properties of a straight line 34 A straight line can be described by an equation of the form y = β 0 + β 1 x β 0 and β 1 are coefficients of the model β 0 is called the y-intercept of the line: the value of y when x = 0 β 1 is called the slope: the amount of change in y for each 1-unit change in x

35 Example data with 15 epidemiologists 35 ID x (Year of working) y (Number of investigation)

36 Approach of SLR 36 The following graph is a scatter plot of the example data with 15 epidemiologists We want to find a line that adequately represents the relationship between X (year of working) and Y (Number of outbreak investigation) y x

37 Algebraic approach 37 The best fitting line is the one with the minimum sum of squared errors The way to find the best-fitting line to minimize the sum of squared errors is called least-squares method Let denote the estimated outcome at based on the fitted Yˆi regression line, β0 1 β Yi =β 0+ β1x i and are the intercept and the slope of the fitted line The error, or residual, is ˆ e i = Y i Yˆ i = Y i X i ( ˆ β + ˆ β X 0 1 i ) The sum of the squares of all such errors is SSE = n i= 1 e 2 i = n i= 1 n 2 ( Y Yˆ ) = ( Y ˆ β ˆ β X i i i= 1 i 0 1 i ) 2

38 Algebraic approach 38 The least-squares estimates ˆβ = 1 n i= 1 ( X i X )( Yi Y ) = n 2 ( X X ) i= 1 ˆ β = Y ˆ β1x 0 i SP SS XY X

39 2 ( X ( Y ( i ( X X i Y i )( ) X Y ) ) Y ) i i Approach of SLR 39 ID x y ( X i X ) ( Y i Y ) ( X i X )( Yi Y ) ( X i X ) Total X =115 /15= 7.67 Y = 281 /15=

40 Algebraic approach 40 SP XY SS X β = = SP SS ˆ1 XY = = = X 2.17 ˆ β ˆ 0 = Y β1x = (2.17)(7.67) = Y = X 2.09

41 Approach of SLR 41 Regression line and equation, graph showing error of data point between observed (10, 40) to expected (10, 23.80) y error y = x x Y =18.73

42 Interpretation of coefficients 42 Y = X β 0 : for epidemiologist with no working experience (X=0), average number of outbreak investigation is 2.17 Less meaningful for continuous variable of X, unless center it at some meaningful value β 1 : for additional one year of working experience, average number of outbreak investigation increase by 2.09

43 SLR for dichotomous variable 43 ID x1 (training) y (Number of investigation) x1 : 0 = no training, 1 = ever attend training

44 SLR for dichotomous variable 44 y x 1

45 SLR for dichotomous variable 45 y y = x x 1

46 Interpretation of coefficients 46 Y = X β 0 : for epidemiologist with no training(x1=0), average number of outbreak investigation is β 1 : for epidemiologist who ever attend training, average number of outbreak investigation increase by (or average number of outbreak investigation is 25.25)

47 Multiple linear regression (MLR) 47 ID X (Year of working) X1 (training) y (Number of investigation) x1 : 0 = no training, 1 = ever attend training

48 Multiple linear regression (MLR) 48 Also used least-squares method to estimate coefficients and draw best fitting line General form Yˆ = β + β X + β X β β 0 is the value of y when all x i = 0 β i is amount of change in y for each 1-unit change in x i adjusted for other covariates i X i

49 Interpretation of coefficients 49 Y = X X1 β 0 : for epidemiologist with no working experience (X=0) and no training(x1=0), average number of outbreak investigation is Less meaningful if there are continuous variables of X in the model, unless center it at some meaningful value β 1 : for additional one year of working experience, average number of outbreak investigation increase by 1.93 after adjusted for ever attend training β 2 : for epidemiologist who ever attend training, average number of outbreak investigation increase by after adjusted for working experience

50 Coefficient of Determination (r 2 or R 2 ) 50 A measure of the impact that X covariate(s) has in explaining the variation of Y 0 r : a perfect prediction 0 : absolutely no association The higher the r 2 value, the more the variation in Y is able to be predicted by X. The higher the r 2, the smaller the errors of prediction Adjusted r 2 : a modification of r 2 that adjusts for the number of covariates in a model

51 Interpretation of r 2 51 Y = X X1 r 2 = % of the variability of number of outbreak investigation is explained by number of year in working experience and ever attend training

52 Hypothesis testing of the model 52 p-value or critical value approach Two parts of testing Significant of overall model (do all covariates together can predict the outcome?) Using ANOVA approach (overall F-test) Significant of each β i (adjusted for other covariates) Using either partial F-test or partial t-test

53 ANOVA table 53 Source Sum of square (SS) df Mean square (MS) F statistics Regression SSR k MSR MSR/MSE Residual (Error) SSE n-k-1 MSE Total SST n-1 2 SSR ( ˆ Y ) 2 SSR ( Y ˆ i Yi ) 2 SST i Y ) = SSR+ SSE = Yi = = ( Y k : number of covariate in the model n : number of observation in the model MSR = SSR/k MSE = SSE/(n-k-1) r 2 = SSR / SST SS y

54 Regression line and error 54 y SSE SSR y = x SST Y = x Note: sum of square (SS) is computed from all y i, this figure just shows only 1 data point of y at (10, 40) as an example

55 Overall F test 55 The ANOVA provides a test for H 0 : β 1 = β 1 = =β i = 0 Test statistics The decision rule is: Reject H 0 if F > F (1 α, k, n k 1) or p-value < α Interpretation: if reject H 0 There was a significant prediction of outcome by covariates taken together (at least one covariate is significant) F k, n k 1 = MSR MSE

56 partial t test 56 It provides a test for H 0 : β i. given other covariates in the model = 0 Test statistics The decision rule is: Reject H 0 if t > t (1 α/2, n k 1) or p-value < α Interpretation: if reject H 0 t n k 1 βi SE There was a significant prediction of outcome by x i adjusted for other covariates in the model = β i

57 partial F test 57 Extra sum of squares We refer to SSR x2 x1 as the EXTRA SUM OF SQUARES due to including X2 in the model after X1 is already there This calculation is done by subtraction, by comparing the SSR from two models, one referred to as the FULL MODEL and the other as the REDUCED MODEL SSR X... X * X... X = SSR p X... X p, X SSR 1 1 X1 Extra SS Full model SS Reduced model SS p We use this extra SSR to calculate a partial F statistic for contribution of that variable in the order we designated

58 partial F test 58 It provides a test for H 0 : β i. given other covariates in the model = 0 Test statistics SSRFull SSR k1 k2 Fk 1 k2, n k1 1 = SSRFull n k1 1 The decision rule is: F > F( 1 α, k k2, n k1 Reject H 0 if or p-value < α 1 1) Reduced Interpretation: if reject H 0 There was a significant prediction of outcome by x i adjusted for other covariates in the model

59 Example command in EpiInfo 59 SLR REGRESS y = x

60 Example output in EpiInfo 60 SLR Partial F test Overall F test

61 Example command in EpiInfo 61 MLR REGRESS y = x x1

62 Example output in EpiInfo 62 MLR Partial F test Overall F test

63 Example command & output in STATA 63 SLR reg y x Partial t test Overall F test

64 Example command & output in STATA 64 MLR reg y x x1 Partial t test Overall F test

65 Logistic regression 65

66 Introduction 66 Dichotomous outcome is very common situation in biology and epidemiology Sick, not sick Dead, alive Cure, no cure What are problems if we use linear regression with dichotomous outcome? For example Outcome is cardiovascular disease (CVD): 0 = Absent, 1= Present Exposure is body mass index (BMI)

67 Introduction 67 A broad class of regression models, collectively known as the generalized linear model, has been developed to address multiple regression with a variety of dependent variable like dichotomies and counts Logistic regression is one generalized linear model for categorical outcome variables Logistic regression allows one to predict a dichotomous (or categorical) outcome from a set of variables that may be continuous, discrete (or categorical), dichotomous, or a mix.

68 BMI and CVD status of 200 subject 68 id bmi cvd id bmi cvd id bmi cvd id bmi cvd id bmi cvd id bmi cvd id bmi cvd id bmi cvd

69 Example data and scatter plot 69 1 CVD BMI

70 Example data and scatter plot 70 Can we model linear regression using least-square method? 1 CVD y = 0.155x BMI

71 Frequency table of BMI level by CVD 71 CVD BMI n Absent Present Proportion 17 to to to to to to to to to to to to to TOTAL

72 Plot of percentage of subjects with CVD in each BMI level 72 % of CVD present BMI level

73 Plot of percentage of subjects with CVD in each BMI level 73 Can we model linear regression using least-square method? % of CVD present y = x BMI level

74 Problem with linear regression 74 Using dichotomous value of y (0 or 1) Meaningless for predicted y Using probability of outcome as y (% of yes) Predicted y can be less than 0 or greater than 1, but probability must lie between 0 and 1 Distribution of data seems to be s-shape (not straight line), resemble a plot of logistic distribution

75 Model a function of y 75 Let Yˆ = β + β X + β X β i X i A denote the right side of equation p or P(y x) denote probability of getting outcome q (or 1-p) denote probability of no outcome 2 For left side of equation p/q or p/(1-p) as y in the model

76 Model a function of y 76 p = 1 p A p = A*(1 p) = A Ap p + Ap= A p (1+ A) = A A p= 1+ A A 1+ A belongs to the interval (0, 1) if A line between 0 and function of A exponential of A e A

77 Logistic function 77 Probability of disease e P( y x)= 1 + e α βx α+ βx 0.0 x

78 Model a function of y 78 p 1 p = β + β X + β X β The right-hand side of equation can take any value between - and The left-hand side of equation can take only value between 0 and (not linear model) 2 2 i X i Solution: take natural log (ln) on both sides Both sides can take any value between - and (as linear model)

79 Logit transformation 79 + e P( y x)= 1 + e α βx α+ βx P( y x) ln 1 P( y x) =β +β 0 i x i logit of P(y x)

80 Advantages of Logit 80 Properties of a linear regression model Logit between - and + Probability (P) constrained between 0 and 1 Directly related to odds of disease p ln 1 p =β +β 0 i x i 1 p p = e β + β 0 i xi e = The exponential function involves the constant with the value of (roughly 2.72).

81 Fitting the equation of data 81 Maximum likelihood Yield values for the unknown parameters which maximize the probability of obtaining the observed set of data Maximum Likelihood Estimates (MLE) for β 0 and β i Likelihood function Express the probability of the observed data as a function of the unknown parameters The resulting estmators are those which agree most closely with the observed data Practically easier to work with log-likelihood: L(β ) Let π ( x i ) = P( y x) n [ l( β )] = { y [ ] + [ ] i lnπ ( xi ) (1 yi )ln 1 ( xi )} L( β ) = ln π i= 1

82 Logistic regression 82 Simple Logistic regression A single covariate Multiple logistic regression Two or more covariate (with or without interactions)

83 Interpreting the estimated regression coefficients 83 The simplest case is when the logistic regression model involves only one covariate (x), and that x takes only two values, 0(unexposed) and 1 (exposed) A logistic regression model for these data would correspond to P( y x1) ln 1 P( y x 1 ) =β +β x 0 1 1

84 Interpreting the estimated regression coefficients 84 For the exposed individuals (X1 = 1) P( y x ln 1 P( y = 1) x1= 1) 1 =β + β For the unexposed individuals (X1 = 0) P( y x1= 0) ln = β0 1 P( y x1= 0) 0 1

85 Interpreting the estimated regression coefficients 85 If we subtract the latter model equation (where X1 = 0) from the former (where X1 = 1) P( y x1= 1) ln 1 P( y x1= 1) D+ D- E+ a b E- c d P( y x1= 0) ln 1 P( y x1= 0) P( y x1= 1) P( y x1= 0) ln = β1 1 P( y x1= 1) 1 P( y x1= 0) a /( a+ b) c /( c+ d) ln = β1 /( ) /( ) b a+ b d c+ d ad = e β 1 bc = ( β + β ) β β = increase in natural log of odds ratio for a one unit increase in x 0 OR= e β 1 1 0

86 Interpreting the estimated regression coefficients 86 β is amount of change in logit for each unit increase in X β is not interpreted directly, instead, e β is interpreted e β is equal to odds ratio; change in odds between a baseline group and a single unit increase in X if e β = 1 then there is no change in odds ratio if e β < 1 then odds ratio decrease if e β > 1 then odds ratio increase e β0 is a baseline odds

87 Example data : CVD 87 Variable Description Value id Subject ID number gender Gender 0=male,1=female age Age in year weight Weigh in kilogram height Height in centimeter bmi Body mass index (kg/m 2 ) hypertension Hypertension status 0=yes,1=no dm Diabetes status 0=yes,1=no alcohol Alcohol drinker 0=yes,1=no smoke Smoker 0=yes,1=no cvd CVD status 0=yes,1=no

88 Simple logistic regression 88 logit ( y ) = *bmi Interpretation of β 1 log odds of CVD increase by 1.61 for a one unit increase in BMI OR = e 1.61 = 5.0 The probability of getting CVD is 5.0 time as likely with an increase of BMI by one unit

89 Simple logistic regression 89 logit ( y ) = * Smoke Interpretation of β 1 log odds of CVD increase by 1.19 for smoker OR = e 1.19 = 3.3 Those who are smoker are 3.3 time as likely to get CVD compare to those who are non-smoker

90 Multiple logistic regression 90 logit ( y) = * smoke+ 0.42* alcohol+ 1.6* bmi Interpretation of β 1 log odds of CVD increase by 0.66 for smoker adjusted for alcohol and BMI OR = e 0.66 = 1.9 Those who are smoker are 1.9 time as likely to get CVD compare to those who are non-smoker adjusted for alcohol and BMI

91 Pseudo R 2 91 In linear regression we have the R 2 as a single agreed upon measure of goodness of fit of the model, the proportion of total variation in dependent variable accounted for by a set of covariates. No single agreed upon index of goodness of fit exists in logistic regression. Instead a number have been defined. These indices are sometimes referred to as Pseudo R 2 s. 2 R L : Cox and Snell Index Nagelkerke Index None of these indices have an interpretation as proportion of variance accounted for as in linear regression. It is important to indicate which index is being used in report.

92 Pseudo R R L : a commonly used index ranges between 0 and 1 can be calculated from the deviance( 2LL) from the null model (no predictors) and the model containing k predictors. Interpreted as the proportion of the null deviance accounted for by the set of predictors R 2 L = Dnull D D null k

93 Hypothesis testing of the model 93 p-value or critical value approach Two parts of testing Significant of overall model (do all covariates together can predict the outcome?) Testing for overall fit of the model Significant of each β i (adjusted for other covariates) Using Wald test

94 Overall fit of the model 94 Measures of model fit and tests of significance for logistic regression are not identical to those in linear regression. In logistic regression, measures of deviance replace the sum of squares of linear regression as the building blocks of measures of fit and statistical tests. These measures can be thought of as analogous to sums of squares, though they do not arise from the same calculations. Each deviance measure in logistic regression is a measure of lack of fit of the data to a logistic model.

95 Overall fit of the model 95 Two measures of deviance are needed the null deviance, D null, which is the analog of SS Y (or SST) in linear regession. D null is a summary number of all the deviance that could potentially be accounted for. It can be thought of as a measure of lack of fit of data to a model containing an intercept but not predictors. It provides a baseline against which to compare prediction from other models that contain at least one predictor. the model deviance from a model containing k predictors, D k ; it is the analog of SSE in linear regression. It is a summary number of all the deviance that remains to be predicted after prediction from a set of k predictors, a measure of lack of fit of the model containing k predictors. If the model containing k predictors fits better than a model containing no predictors, then D k < D null. This is the same idea as in linear regression: if a set of predictors in linear regression provides prediction, then SSE < SST

96 Overall fit of the model 96 The statistical tests built on deviances are referred to collectively as likelihood ratio tests because the deviance measures are derived from ratios of maximum likelihoods under different models. Standard notation for deviance : 2LL or 2 log likelihood Likelihood ratio test (LR) H 0 : β 1 = β 1 = =β i = 0 A likelihood ratio test = D null D k, which follows a distribution with df = k reject H 0 if observed is greater than 100(1-α) percentile of distribution with k df 2 2 χ χ 2 χ

97 Testing of β 97 If an acceptable model is found, the statistical significance of each of the coefficients is evaluated using the Wald test Using chi-square test with 1 df W i = β S 2 i 2 β i Using z test W i = βi S β i

98 Wald test 98 It provides a test for H 0 : β i. given other covariates in the model = 0 Test statistics: chi-square or z test The decision rule is: Reject H 0 if 2 χ Wald s (1, df=k ), or Wald s z > Z (1 α/2) p-value < α 2 > χ α Interpretation: if reject H 0 There was a significant prediction of outcome by x i adjusted for other covariates in the model

99 Model building 99 Depends on study objectives single hypothesis testing : estimate the effect of a given variable adjusted for all potential available confounders exploratory study : identify a set of variables independently associated to the outcome adjusted for all potential available confounders to predict : predict the outcome with the least varieble possible (parsimony principle)

100 Model building strategy 100 Variable selection Approach Hierarchical or sequential regression: investigator s judgement (preferred) Best subset: computer s judgement (not recommend) Exclusion of least significant variables base on statitical test and no important variation of coefficient until obtaining a satisfactory model Add and test interaction terms Final model with interaction term if any Check model fit

101 Variable selection 101 what is the hypothesis, what is (are) the variable(s) of interest Define variables to keep in the model (forced) what are the confounding variables Literature review Confounding effect in the dataset (bivariate analysis) biologic pathway and plausibilty statistical (p-value) Variable with p-value < 0.2 Form of continuous variables Original form as continuous Categorized variable and coding

102 Hierarchical regression 102 Assess whether a set of m predictors contribute significant prediction over and above a set of k predictors likelihood ratio (LR) test Deviance are computed for the k predictor model, Dk, and the (m + k) predictor model, Dm+k The difference between these deviances is an LR test for the significance of contribution of the set of m predictors over and above the set of k predictors, with df = m The LR tests are used to compare models that are nested : all the predictors in the smaller (reduced) model are included among the predictors in the larger (full) model. Backward elimination or forward selection approach

103 Likelihood ratio statistic 103 Compares two nested models Log(odds) = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 (model 1) Log(odds) = β 0 + β 1 x 1 + β 2 x 2 (model 2) LR statistic -2LL model 2 minus -2LL model 1 LR statistic is a χ 2 with df = number of extra parameters in model

104 Example 104 Full Model: smoke, alcohol, bmi Deviance (-2LL): Reduced model: smoke, bmi Deviance (-2LL): H 0 : reduced model contribute significant prediction over full model Test statistics LR test = = 0.36 df = 3-2 = 1 2 χ Critical value of (. 95, df= 1) = 3.84 Reject the H 0 Conclusion: alcohol does no contribute significant prediction in the model

105 Interaction term 105 If 2 covariates are x1 and x2 Interaction is include in the model as x1*x2 Assessing interaction use the same strategy of LR test Full model : x1, x2, x1*x2 Reduced model : x1, x2 df = 3-2 = 1

106 Assumption 106 No multicollinearity among covariates Linear relationships between continuous variables and a logit No outliers among errors

107 Check model fit 107 Summary measure of Goodness-of-Fit Pearson Chi-square statistics and deviance approach Hosmer-Lemeshow test Area under ROC curve Model diagnostics Check whether the model violate the assumptions

108 Example command in EpiInfo 108 Simple logistic regression LOGISTIC cvd = bmi

109 Example output in EpiInfo 109 Simple logistic regression Wald test Deviance (-2LL) LR test (to null)

110 Example command in EpiInfo 110 Simple logistic regression LOGISTIC cvd = smoke

111 Example output in EpiInfo 111 Simple logistic regression Wald test Deviance (-2LL) LR test (to null)

112 Example command in EpiInfo 112 Multiple logistic regression LOGISTIC cvd = alcohol bmi smoke

113 Example output in EpiInfo 113 Multiple logistic regression Wald test Deviance (-2LL) LR test (to null)

114 Example command & output in STATA 114 Simple logistic regression: compute coefficients logit cvd bmi LR test (to null) LL Wald test

115 Example command & output in STATA 115 Simple logistic regression: compute OR logit cvd bmi, or Logistic cvd bmi LR test (to null) LL Wald test

116 Example command & output in STATA 116 Simple logistic regression: compute coefficients logit cvd smoke LR test (to null) LL Wald test

117 Example command & output in STATA 117 Simple logistic regression: compute OR logit cvd smoke, or logistic cvd smoke LR test (to null) LL Wald test

118 Example command & output in STATA 118 Multiple logistic regression: compute coefficients logit cvd smoke alcohol bmi LR test (to null) LL Wald test

119 Example command & output in STATA 119 Multiple logistic regression: compute OR logit cvd smoke alcohol bmi, or logistic cvd smoke alcohol bmi LR test (to null) LL Wald test

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