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From t-test to … multilevel analyses

Learn how to interpret and apply sophisticated statistical analyses, such as linear regression, GLM, GEE, and GLMM, using software like SPSS, Stata, R, and MLwiN. Explore multilevel models and handle missing data in cortisol studies.

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From t-test to … multilevel analyses

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  1. From t-test to … multilevel analyses Stein Atle Lie

  2. Outline • Pared t-test (Mean and standard deviation) • Two-group t-test (Mean and standard deviations) • Linear regression • GLM (general linear models) • GEE (general estimation equations) • GLMM (general linear mixed model) • … • SPSS, Stata, R, MLwiN, gllamm (Stata)

  3. Multilevel models • “Same thing – many names”: • Generalized estimation equations • Random effects models • Random intercept and random slope models • Mixed effects models • Variance component models • Frailty models (in survival analyses) • Latent variables

  4. Cortisol data – missing data

  5. Objective • Take the general thinking from simple statistical methods into more sophisticated data-structures and statistical analyses • Focus on the interpretation of the results with respect to those found in basic statistical methods

  6. Multilevel data Types of data: • Repeated measures for the same individual • The same measure is repeated several times on the same individual • Several observers have measured the same individual • Several different measures for the same individual • Related observations (siblings, families, …) • A categorical variable with ”many” levels (multicenter data, hospitals, clinics, …) • Panel data

  7. Null hypotheses • In ordinary statistics (using both pared and two‑sample t-tests) we define a null hypothesis. H0: m1 = m2 • We assume that mean from group (or measure) 1 is equal to the mean from group (or measure) 2. • Alternatively H0: D = m1-m2 = 0

  8. p-value • Definition: • “If our null-hypothesis is true - what is the probability to observe the data* that we did?” * And hence the mean, t-statistic, etc…

  9. p-value • We assume that our null-hypothesis is true (m0=0 or m1-m2=0) • We observe our data • Mean value etc. • Under the assumption of normal distributed data p-value • The p-value is the probability to observe our data (or something more extreme) under the given assumptions m0

  10. Pared t-test • The straightforward way to analyze two repeated measures is a pared t-test. • Measure at time1 or location1 (e.g. Data1) is directly compared to measure at time2 or location2 (e.g. Data2) • Is the difference between Data1 and Data2 (Diff = Data1-Data2) unlike 0?

  11. Pared t-test (n=10) PASW: T-TEST PAIRS=Data1 WITH Data2 (PAIRED).

  12. Pared t-test • The pared t-test will only be performed for complete (balanced) data. • What happens if we delete two observations from data2? • (Only 8 complete pairs remain)

  13. Pared t-test (n=8) PASW: T-TEST PAIRS=Data1 WITH Data2 (PAIRED). Excel

  14. Two group t-test • If we now consider the data from time1 and time2 (or location1 and location2) to be independent (even if their not) and use a two group t-test on the full dataset, 2*10 observations

  15. Two group t-test (n=20 [10+10]) PASW: T-TEST GROUPS=Grp(1 2) /VARIABLES=Data.

  16. Two group t-test • Observe that mean for Grp1 and Grp2 is equal to mean for Data1 and Data2 • And that the mean difference is also equal • The difference between pared t-test and two group t-test lies in the • Variance - and the number of observations • and therefore in the standard deviation and standard error • and hence in the p-value and confidence intervals

  17. Two group t-test • The two group t-test are performed on all available data. • What happens if we delete two observations from Grp2? • (Only 8 complete pairs remain - but 18 observations remain!)

  18. Two group t-test (n=18 [10+8]) PASW: T-TEST GROUPS=Grp(1 2) /VARIABLES=Data.

  19. Two group t-test (s1=s2) s1 s2 m1 m2 D

  20. Two group t-test (s1=s2) s1 s2

  21. ANOVA (Analysis of variance (s1=s2=s3) s3 s1 s2

  22. Linear regression • If we now perform an ordinary linear regression with the data as outcome (dependent variable) and the group variable (Grp=1 and 2) as independent variable • the coefficient for group is identical to the mean difference • and the standard error, t-statistic, and p‑value are identical to those found in a two‑group t‑test

  23. Linear regression (n=20) Stata: . regress data grp Source | SS df MS Number of obs = 20 -------------+------------------------------ F( 1, 18) = 1.38 Model | 21.0124998 1 21.0124998 Prob > F = 0.2554 Residual | 274.01701 18 15.2231672 R-squared = 0.0712 -------------+------------------------------ Adj R-squared = 0.0196 Total | 295.02951 19 15.5278689 Root MSE = 3.9017 ------------------------------------------------------------------------------ data | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- grp | 2.05 1.744888 1.17 0.255 -1.615873 5.715873 _cons | 5.33 2.75891 1.93 0.069 -.4662545 11.12625 ------------------------------------------------------------------------------

  24. Linear regression • Now exchange the independent variable for group (Grp=1 and 2) with a dummy variable (dummy=0 for grp=1 and dummy=1 for grp=2) • the coefficient for the dummy is equal to the coefficient for grp (the mean difference) • and the coefficient for the constant term is equal to the mean for grp1 (the standard error is not!)

  25. Linear regression (n=20) Stata: . regress data dummy Source | SS df MS Number of obs = 20 -------------+------------------------------ F( 1, 18) = 1.38 Model | 21.0124998 1 21.0124998 Prob > F = 0.2554 Residual | 274.01701 18 15.2231672 R-squared = 0.0712 -------------+------------------------------ Adj R-squared = 0.0196 Total | 295.02951 19 15.5278689 Root MSE = 3.9017 ------------------------------------------------------------------------------ data | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- dummy | 2.05 1.744888 1.17 0.255 -1.615873 5.715873 _cons | 7.38 1.233822 5.98 0.000 4.787836 9.972164 ------------------------------------------------------------------------------

  26. Linear models in Stata • In ordinary linear models (regress and glm) in Stata one may add an option for clustered data – to obtain standard errors adjusted for intragroup correlation • This is ideal when you want to adjust for clustered data, but are not interested in the correlation within or between groups • And - you will still have the population effects!!

  27. Linear regression (n=20) Stata: . regress data dummy, cluster(id) Linear regression Number of obs = 20 F( 1, 9) = 2.64 Prob > F = 0.1388 R-squared = 0.0712 Root MSE = 3.9017 (Std. Err. adjusted for 10 clusters in id) ------------------------------------------------------------------------------ | Robust data | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- dummy | 2.05 1.262145 1.62 0.139 -.8051699 4.90517 _cons | 7.38 1.224847 6.03 0.000 4.609204 10.1508 ------------------------------------------------------------------------------

  28. Linear models in Stata • Thus, we now have an alternative to the pared t‑test. The mean difference is identical to that obtained from the pared t‑test, and the standard errors (and p-values) are adjusted for intragroup correlation • As an alternative we may use the program gllamm (Generalized Linear Latent And Mixed Models) in Stata • http://www.gllamm.org/

  29. gllamm (n=20) gllamm (Stata): . gllamm data dummy, i(id) number of level 1 units = 20 number of level 2 units = 10 ------------------------------------------------------------------------------ data | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- dummy | 2.05 1.167852 1.76 0.079 -.2389486 4.338949 _cons | 7.379808 1.172819 6.29 0.000 5.081124 9.678492 ------------------------------------------------------------------------------ Variance at level 1 6.8193955 (3.0174853) Variances and covariances of random effects ------------------------------------------------------------------------------ level 2 (id) var(1): 6.8114516 (4.5613185)

  30. Linear models in Stata • If we now delete two of the observations in Grp2 • We then have coefficients (“mean differences”) calculated based on all (n=18) data • and standard errors corrected for intragroup correlation - using the commands <regress>, <glm> or <gllamm>

  31. Linear regression (n=18) Stata: . regress data dummy, cluster(id) Linear regression Number of obs = 18 F( 1, 9) = 1.63 Prob > F = 0.2332 R-squared = 0.0587 Root MSE = 4.1303 (Std. Err. adjusted for 10 clusters in id) ------------------------------------------------------------------------------ | Robust data | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- dummy | 1.9575 1.531486 1.28 0.233 -1.506963 5.421963 _cons | 7.38 1.228869 6.01 0.000 4.600105 10.1599 ------------------------------------------------------------------------------

  32. gllamm (n=18) gllamm (Stata): . gllamm data dummy, i(id) number of level 1 units = 18 number of level 2 units = 10 log likelihood = -48.538837 ------------------------------------------------------------------------------ data | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- dummy | 2.458305 1.253552 1.96 0.050 .0013882 4.915223 _cons | 7.357426 1.232548 5.97 0.000 4.941677 9.773176 ------------------------------------------------------------------------------ Variance at level 1 6.4041537 (3.3485133) level 2 (id) var(1): 8.7561818 (5.1671805)

  33. Intra class correlation (ICC) Variance at level 1 6.4041537 (3.3485133) level 2 (id) var(1): 8.7561818 (5.1671805) • The total variance is hence • 6.4041 + 8.7561= 15.1603 • (and the standard deviation is hence 3.8936) • The proportion of variance attributed to level 2 is therefore • ICC = 8.7561/15.1603 = 0.578

  34. Linear regression • Ordinary linear regression • Assumes data is Normal and i.i.d. (identical independent distributed)

  35. Linear regression b1 residual Y Regression line: y = b0 + b1·x (x1,y1) (xn,yn) (xi,yi) b0 Height * Weight Kortisol * Months Kortisol * Time X

  36. Linear regression • Assumptions: 1) y1, y2,…, yn are independent normal distributed 2) The expectation of Yi is: E(Yi) = b0+ b1·xi(linear relation between X and Y) 3) The variance of Yi is: var(Yi) = s2(equal variance for ALL values of X)

  37. Linear regression • Assumptions - Residuals (ei): yi = a + b·xi + ei 1) e1, e2,…, en are independent normal distributed 2) The expectation of ei is: E(ei) = 0 3) The variance of ei is: var(Yi) = s2

  38. ^ yi=a+b·xi ^ (yi-yi)2 ^ _ y (xi,yi) _ x Y Regression What is the ”best” a and b? Least squares method (xi,yi) residual (e) residual (e) X

  39. Regression • Least squares method: • We wish that the sum of squares (The distance from all points to the line [the residuals]; squared) is as least as possible – we whish to find the minimum

  40. Regression • The least squares method: • The solution is:

  41. Regression • The maximum likelihood method: • Assumptions: 1) y1, y2,…, yn are random (independent), normal-distributed observations, i.i.d. 2) Expectation for Yi is: E(Yi) = a + b·xi 3) Variance for Yi is: var(Yi) = s2 f(y) maximized v.r.t. a and b. (The likelihood-function) This is the same as finding the minimum of For simple linear regression the least squares method and the maximum likelihood method are equal!

  42. ^ _ y (xi,yi) _ x Y Regression The maximum likelihood method ”The probability that the line fits the observed points” residual (e) (xi,yi) X

  43. Ordinary linear regression • The formula for an ordinary regression can be expressed as: yi = b0 + b1·xi + ei ei ~N(0, se2)

  44. 100 90 80 Vekt i kg (Y) 70 60 Kvinner Menn 50 150 160 170 180 190 200 210 Høyde i cm (X) Interpretation of coefficients Y = - 97.6 + 0.96*X Y = a + b*X Det vil si: a = -97.6 og b=0.96

  45. Interpretation of coefficients Y = - 85.0 + 0.91*X1 - 1.86*X2 } = 1.86 kg

  46. Random intercept model b1 Y Regression lines: yij = b0 + b1·xij+vij (x11,y11) (xnp,ynp) b0+uj (xij,yij) su se X

  47. Random intercept model • For a random intercept model, we can express the regression line(s) - and the variance components as yij = b0 + b1·xij + vij vij = uj + eij eij ~N(0, se2) (individual) uj ~N(0, su2) (group)

  48. Random intercept model • Alternatively we may express the formulas, for the simple variance component model, in terms of random intercepts: yij = b0j + b1·xij + eij b0j = b0 + uj eij ~N(0, se2) (individual) uj ~N(0, su2) (group)

  49. Random slope model • For a random slope model (the intercepts are equal), we can express the regression line(s) and the variance components as yij = b0 + b1j·xij + eij b1j = b1+ wj eij ~N(0, se2) (individual) wj ~N(0, sw2) (group)

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