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Summer School

Summer School. Week 2. Contents. Logistic regression refresher Some familiar + some less familiar polytomous models 1PL/2PL in Stata and R PCM/RSM/GRM in R Link IRT to CFA/UV in Mplus DIF/MIMIC in Mplus. Types of outcome. Two categories Binary / dichotomous Ordered

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Summer School

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  1. Summer School Week 2

  2. Contents • Logistic regression refresher • Some familiar + some less familiar polytomous models • 1PL/2PL in Stata and R • PCM/RSM/GRM in R • Link IRT to CFA/UV in Mplus • DIF/MIMIC in Mplus

  3. Types of outcome • Two categories • Binary / dichotomous • Ordered • e.g. low birthweight (< 2500g), height > 6ft, age > 70 • Unordered • e.g. gender, car-ownership, disease status • Presence of ordering is unimportant for binaries

  4. Types of outcome • 3+ categories • Polytomous • Ordered (ordinal) • Age (<30,30-40,41+) • “Likert” items (str disagree, disagree, …, str agree) • Unordered (nominal) • Ethnicity (white/black/asian/other) • Pet ownership (none/cat/dog/budgie/goat)

  5. Modelling options (LogR/IRT)

  6. Binary Logistic Regression

  7. Binary Logistic Regression Probability of a positive response / outcome given a covariate Intercept Regression coefficient

  8. Binary Logistic Regression Probability of a negative response

  9. Logit link function • Probabilities only in range [0,1] • Logit transformation is cts in range (–inf,inf) • Logit is linear in covariates

  10. Simple example – cts predictor • Relationship between birthweight and head circumference (at birth) • Exposure • birthweight (standardized) variable | mean sd ----------+------------------ bwt | 3381.5g 580.8g -----------------------------

  11. Simple example – cts predictor • Outcome • Head-circumference ≥ 53cm headcirc | Freq. % ----------+----------------- 0 | 8,898 84.4% 1 | 1,651 15.7% ----------+----------------- Total | 10,549

  12. Simple example – cts predictor The raw data – doesn’t show much

  13. Simple example – cts predictor Logistic regression models the probabilities (here shown for deciles of bwt) | bwt_z_grp headcirc | 0 1 2 3 4 | -----------+-------------------------------------------------------+-- 0 | 1,006 993 1,050 946 1,024 | | 99.80 98.12 97.95 96.04 93.35 | -----------+-------------------------------------------------------+-- 1 | 2 19 22 39 73 | | 0.20 1.88 2.05 3.96 6.65 | -----------+-------------------------------------------------------+-- headcirc | 5 6 7 8 9 | Total -----------+-------------------------------------------------------+---------- 0 | 931 922 856 688 381 | 8,797 | 89.95 84.98 81.84 66.67 35.94 | 84.33 -----------+-------------------------------------------------------+---------- 1 | 104 163 190 344 679 | 1,635 | 10.05 15.02 18.16 33.33 64.06 |15.67 -----------+-------------------------------------------------------+----------

  14. Simple example – cts predictor Increasing, non-linear relationship

  15. Simple example – cts predictor Logistic regression Number of obs = 10432 LR chi2(1) = 2577.30 Prob > chi2 = 0.0000 Log likelihood = -3240.9881 Pseudo R2 = 0.2845 ------------------------------------------------------------------------------ headcirc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- bwt_z | 7.431853 .378579 39.38 0.000 6.72569 8.212159 ------------------------------------------------------------------------------ Or in less familiar log-odds format ------------------------------------------------------------------------------ headcirc | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- bwt_z | 2.005775 .0509401 39.38 0.000 1.905935 2.105616 _cons | -2.592993 .0474003 -54.70 0.000 -2.685896 -2.50009 ------------------------------------------------------------------------------

  16. Simple example – cts predictor Fitted model – logit scale

  17. Simple example – cts predictor Fitted model – logit scale Cons = -2.59 Slope = 2.00

  18. Simple example – cts predictor But also…a logit of zero represents point at which both levels of outcome are equally likely

  19. Simple example – cts predictor Fitted model – probability scale

  20. Simple example – cts predictor Fitted model – probability scale Point at which curve changes direction

  21. Simple example – cts predictor Observed and fitted values (within deciles of bwt)

  22. LogR cts predictor - summary • Logit is linearly related to covariate • Gradient gives strength of association • Intercept is related to prevalence of outcome • + seldom used • Non-linear (S-shaped) relationship between • probabilities and covariate • Steepness of linear-section infers • strength of association • Point at which curve changes direction is where • P(u=1|X) = P(u=0|X) can be thought of as the • location + isrelated to prevalence of outcome

  23. LogR – binary predictor • Define binary predictor: bwt ≥ 8lb • 32% of the sample had a birthweight of 8lb+ • Same outcome • Head circumference > 53cm • Does being 8lb+ at birth increase the chance of you being born with a larger head?

  24. Association can be cross-tabbed headcirc bwt_8lb | 0 1 | Total -----------+----------------------+---------- 0 | 6,704 384 | 7,088 | 94.58 5.42 | 100.00 -----------+----------------------+---------- 1 | 2,093 1,251 | 3,344 | 62.59 37.41 | 100.00 -----------+----------------------+---------- Total | 8,797 1,635 | 10,432 | 84.33 15.67 | 100.00

  25. Association can be cross-tabbed headcirc bwt_8lb | 0 1 | Total -----------+----------------------+---------- 0 | 6,704384 | 7,088 | 94.58 5.42 | 100.00 -----------+----------------------+---------- 1 | 2,0931,251 | 3,344 | 62.59 37.41 | 100.00 -----------+----------------------+---------- Total | 8,797 1,635 | 10,432 | 84.33 15.67 | 100.00 Familiar with (6704*1251)/(2093*384) = 10.43 = odds-ratio

  26. Association can be cross-tabbed headcirc bwt_8lb | 0 1 | Total -----------+----------------------+---------- 0 | 6,704384 | 7,088 | 94.58 5.42 | 100.00 -----------+----------------------+---------- 1 | 2,0931,251 | 3,344 | 62.59 37.41 | 100.00 -----------+----------------------+---------- Total | 8,797 1,635 | 10,432 | 84.33 15.67 | 100.00 Familiar with (6704*1251)/(2093*384) = 10.43 = odds-ratio However ln[(6704*1251)/(2093*384)] = 2.345 = log odds-ratio

  27. Association can be cross-tabbed headcirc bwt_8lb | 0 1 | Total -----------+----------------------+---------- 0 | 6,704 384 | 7,088 | 94.58 5.42 | 100.00 -----------+----------------------+---------- 1 | 2,093 1,251 | 3,344 | 62.59 37.41 | 100.00 -----------+----------------------+---------- Total | 8,797 1,635 | 10,432 | 84.33 15.67 | 100.00 Familiar with (6704*1251)/(2093*384) = 10.43 = odds-ratio However ln[(6704*1251)/(2093*384)] = 2.345 = log odds-ratio and ln[(384)/(6704)]= ln(0.057) = -2.86 = intercept on logit scale

  28. Logit output (from Stata) . logit headcirc bwt_8lb Logistic regression Number of obs = 10432 LR chi2(1) = 1651.89 Prob > chi2 = 0.0000 Log likelihood = -3703.6925 Pseudo R2 = 0.1823 ------------------------------------------------------------------------------ headcirc | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- bwt_8lb | 2.345162 .063486 36.94 0.000 2.220732 2.469592 _cons | -2.859817 .0524722 -54.50 0.000 -2.962661 -2.756974 ------------------------------------------------------------------------------

  29. What lovely output figures! There is still an assumed s-shape on probability scale although the curve is not apparent Linear relationship in logit space

  30. What lovely output figures! Intercept = -2.86 Slope = 2.35 There is still an assumed s-shape on probability scale although the curve is not apparent Linear relationship in logit space

  31. LogR binary predictor - summary • The same maths/assumptions underlie the models with a binary predictor • Estimation is simpler – can be done from crosstab rather than needing ML • Regression estimates relate to linear relationship on logit scale

  32. Multinomial Logistic Regression

  33. Multinomial Logistic Regression • Typically used for non-ordinal (nominal) outcomes • Can be used for ordered data (some information is ignored) • 3+ outcome levels • Adding another level adds another set of parameters so more than 4 or 5 levels can be unwieldy

  34. Multinomial Logistic Regression where c0 = α0 = 0 Here the probabilities are obtained by a “divide-by-total” procedure

  35. Examples • Outcome: head-circumference • 4 roughly equal groups (quartiles) • Ordering will be ignored headcirc4 | Freq. Percent ------------+--------------------------- <= 49cm | 2,574 24.4% 49.1–50.7cm | 2,655 25.2% 50.8–51.9cm | 2,260 21.4% 52+ cm | 3,060 29.0% ------------+--------------------------- Total | 10,549 100.00 • Exposure 1: birthweight of 8lb or more • Exposure 2: standardized birthweight

  36. Exposure 1: bwt > 8lb • 32% of the sample had a birthweight of 8lb+ • Does being 8lb+ at birth increase the chance of you being born with a larger head • Unlike the logistic model we are concerned with three probabilities • P(headcirc = 49.1 – 50.7cm) • P(headcirc = 50.8 – 51.9cm) • P(headcirc = 52+cm) • Each is referenced against the “negative response” i.e. that headcirc <= 49cm

  37. Exposure 1: bwt > 8lb . mlogit headcirc4 bwt_8lb, baseoutcome(0) Multinomial logistic regression -------------------------------------------------------------- headcirc4 | Coef. SE z P>|z| [95% CI] -------------+------------------------------------------------ 1 | bwt_8lb | 1.56 .135 11.53 0.000 1.30 1.83 _cons | -.07 .029 -2.30 0.022 -0.12 -0.01 -------------+------------------------------------------------ 2 | bwt_8lb | 3.09 .129 23.98 0.000 2.84 3.34 _cons | -.58 .034 -17.33 0.000 -0.65 -0.52 -------------+------------------------------------------------ 3 | bwt_8lb | 4.39 .127 34.43 0.000 4.14 4.64 _cons | -.99 .039 -25.56 0.000 -1.06 -0.92 -------------------------------------------------------------- (headcirc4==0 is the base outcome) 3 sets of results Each is reference against the “baseline” group, i.e. <=49cm

  38. Exposure 1: bwt > 8lb . mlogit headcirc4 bwt_8lb, baseoutcome(0) Multinomial logistic regression --------------------------------- headcirc4 | Coef. (SE) -------------+------------------- 1 | bwt_8lb | 1.56 (.135) _cons | -.07 (.029) -------------+------------------- 2 | bwt_8lb | 3.09 (.129) _cons | -.58 (.034) -------------+------------------- 3 | bwt_8lb | 4.39 (.127) _cons | -.99 (.039) --------------------------------- (headcirc4==0 is the base outcome) Logistic regression ---------------------------------- head_1 | Coef. Std. Err. --------+------------------------- bwt_8lb | 1.56099 .1353772 _cons | -.0664822 .0289287 ---------------------------------- Logistic regression ---------------------------------- head_2 | Coef. Std. Err. --------+------------------------- bwt_8lb | 3.088329 .1287576 _cons | -.5822197 .0335953 ---------------------------------- Logistic regression ---------------------------------- head_3 | Coef. Std. Err. --------+------------------------- bwt_8lb | 4.389338 .127473 _cons | -.9862376 .0385892 ----------------------------------

  39. Exposure 1: bwt > 8lb . mlogit headcirc4 bwt_8lb, baseoutcome(0) Multinomial logistic regression --------------------------------- headcirc4 | Coef. (SE) -------------+------------------- 1 | bwt_8lb | 1.56 (.135) _cons | -.07 (.029) -------------+------------------- 2 | bwt_8lb | 3.09 (.129) _cons | -.58 (.034) -------------+------------------- 3 | bwt_8lb | 4.39 (.127) _cons | -.99 (.039) --------------------------------- (headcirc4==0 is the base outcome) Logistic regression ---------------------------------- head_1 | Coef. Std. Err. --------+------------------------- bwt_8lb | 1.56099 .1353772 _cons | -.0664822 .0289287 ---------------------------------- Logistic regression ---------------------------------- head_2 | Coef. Std. Err. --------+------------------------- bwt_8lb | 3.088329 .1287576 _cons | -.5822197 .0335953 ---------------------------------- Logistic regression ---------------------------------- head_3 | Coef. Std. Err. --------+------------------------- bwt_8lb | 4.389338 .127473 _cons | -.9862376 .0385892 ----------------------------------

  40. Exposure 1: bwt > 8lb • For a categorical exposure, a multinomial logistic model fitted over 4 outcome levels gives the same estimates as 3 logistic models, i.e. Logit(0v1) Multinomial(0v1,0v2,0v3) ≡ Logit(0v2) Logit(0v3) • In this instance, the single model is merely more convenient and allows the testing of equality constraints

  41. Exposure 2: Continuous bwt • Using standardized birthweight we are interesting in how the probability of having a larger head, i.e. • P(headcirc = 49.1 – 50.7cm) • P(headcirc = 50.8 – 51.9cm) • P(headcirc = 52+cm) increases as birthweight increases As with the binary logistic models, estimates will reflect • A change in log-odds per SD change in birthweight • The gradient or slope when in the logit scale

  42. Exposure 2: Continuous bwt mlogit headcirc4 bwt_z, baseoutcome(0) Multinomial logistic regression -------------------------------------------------------------- headcirc4 | Coef. SE z P>|z| [95% CI] -------------+------------------------------------------------ 1 | bwt_z | 2.10 .063 33.11 0.000 1.97 2.22 _cons | 1.06 .044 23.85 0.000 0.97 1.14 -------------+------------------------------------------------ 2 | bwt_z | 3.52 .078 44.89 0.000 3.37 3.68 _cons | 0.78 .046 16.95 0.000 0.69 0.87 -------------+------------------------------------------------ 3 | bwt_z | 4.88 .086 56.90 0.000 4.72 5.05 _cons | 0.33 .051 6.51 0.000 0.23 0.43 -------------------------------------------------------------- (headcirc4==0 is the base outcome)

  43. Exposure 2: Continuous bwt Logistic regression ------------------------------------- head_1 | Coef. Std. Err. -------+----------------------------- bwt_z | 2.093789 .0650987 _cons | 1.058445 .0447811 ------------------------------------- Logistic regression ------------------------------------- head_2 | Coef. Std. Err. -------+----------------------------- bwt_z | 3.355041 .0959539 _cons | .6853272 .0464858 ------------------------------------- Logistic regression ------------------------------------- head_3 | Coef. Std. Err. -------+----------------------------- bwt_z | 3.823597 .1065283 _cons | .3129028 .0492469 ------------------------------------- Multinomial logistic regression ------------------------------ headcirc4 | Coef. (SE) -------------+---------------- 1 | bwt_z | 2.10 (.063) _cons | 1.06 (.044) -------------+---------------- 2 | bwt_z | 3.52 (.078) _cons | 0.78 (.046) -------------+---------------- 3 | bwt_z | 4.88 (.086) _cons | 0.33 (.051) ------------------------------ (headcirc4==0 is the base outcome) No longer identical

  44. Exposure 2: Continuous bwt Outcome level 2 = [49.1 – 50.7] Intercept = 1.06 Slope = 2.10 Shallowest Outcome level 3 = [50.8 – 51.9] Intercept = 0.78 Slope = 3.52 Outcome level 4 = [52.0 –] Intercept = 0.33 Slope = 4.88 Steepest Risk of being in outcome level 4 increases most sharply as bwt increases

  45. Can plot probabilities for all 4 levels

  46. Or altogether on one graph….

  47. Or altogether on one graph….

  48. Ordinal Logistic Models

  49. Ordinal Logistic Models • When applicable, it is useful to favour ordinal models over multinomial models • If outcome levels are increasing, e.g. in severity of a condition or agreement with a statement, we expect the model parameters to behave in a certain way • The typical approach is to fit ordinal models with constraints resulting in greater parsimony (less parameters)

  50. Contrasting among response categories - some alternative models For a 4-level outcome there are three comparisons to be made Model 1 – that used in the multinomial logistic model Model 2 – used with the proportional-odds ordinal model Model 3 – adjacent category model

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