Modeling the Mean: Response Profiles v. Parametric Curves

Transcription

1 Modeling the Mean: Response Profiles v. Parametric Curves Jamie Monogan University of Georgia Escuela de Invierno en Métodos y Análisis de Datos Universidad Católica del Uruguay Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 1 / 18

2 Objectives By the end of this meeting, participants should be able to: Describe the logic and interpretation of analysis of response profiles with a linear model. Test for a group time interaction, a time effect, and a group effect in an analysis of response profiles model. Estimate a linear model that parametrically models time. Estimate a linear model that includes a linear spline of time. Test for a group time interaction in a higher-order polynomial or spline model. Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 2 / 18

3 An Aside on Vocabulary A general linear model is a comprehensive way of thinking about linear models. Among other models, the framework incorporates ANOVA, ANCOVA, linear regression, and multivariate versions of these models with multiple outcome variables. Common estimators include, but are not limited to, least squares and best linear unbiased predictions (BLUP, more on this later). Errors are usually assumed to follow a normal or multivariate normal distribution. A generalized linear model (GLM) is a comprehensive way of thinking about linear models. Among other models, the framework incorporates linear regression, logistic regression, Poisson regression, and ordered choice models. Common estimators include, but are not limited to, maximum likelihood and Bayesian statistics. The conditional distribution of the outcome variable could come from any distribution in the exponential family. Link functions allow for a linear function of a transformed version of the term of interest. Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 3 / 18

4 Response Profiles The Logic of Analysis of Response Profiles Consider: Are you working with experimental or quasi-experimental data? (The chapters are written with this more in mind.) Are you working with pure field data? Either way, we need to account for time dynamics. What is the mean value of a response variable contingent on time and group? (Group number not limited.) We could estimate G n separate means for G groups and n waves. (An ANOVA setup.) Instead, we estimate G n parameters in a reference group parameterization of a general linear model. This is mathematically equivalent to ANOVA. Linear model framework lends itself to more extensions. Note: this method requires a balanced design, though it can include individuals with some missing observations. In other words, if no individual has a missing measurement, n i = n. Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 4 / 18

5 Response Profiles In General: Inference in MLE Wald test One condition: Normal or χ 2 (1) Confidence intervals r conditions: χ 2 (r) Likelihood ratio test Nested models r restrictions G 2 = 2(ˆl full ˆl reduced ) χ 2 (r) How do these compare to post hoc t-ratios or block F -tests from linear regression? Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 5 / 18

6 Response Profiles Hypotheses Concerning Response Profiles 1 Are the mean response profiles similar in the groups, in the sense that the mean response profiles are parallel? See (a). This is the big advantage of studying longitudinal data. If there are no trajectory differences, then we might ask the next two questions. 2 Assuming the population mean response profiles are parallel, are the means constant over time, in the sense that the man response profiles are flat? See (b). 3 Assuming that the population mean response profiles are parallel, are they also at the same level in the sense that the mean response profiles for the groups coincide? See (c). Figure source: Fitzmaurice, Laird, & Ware 2011, Figure 5.2. Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 6 / 18

7 Response Profiles Reference Group Parameterization Recall: Two features of linear models Nominal Inputs: reference group Interaction Effects: multiply values of a variable Response Profile Model (G = 2 & n = 3) y ij = β 0 + β 1 D i + β 2 T {2} ij + β 3 T {3} ij + β 4 D i T {2} ij + β 5 D i T {3} ij + ɛ ij Mean of y ij when D i = 1 and time= T {2} E(y ij X ij ) = β 0 + β β β β β E(y ij X ij ) = β 0 + β 1 + β 2 + β 4 Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 7 / 18

8 Response Profiles Reference Group Parameterization Continued Response Profile Model (G = 2 & n = 3) y it = β 0 + β 1 D i + β 2 T {2} ij + β 3 T {3} ij + β 4 D i T {2} ij + β 5 D i T {3} ij + ɛ ij Mean of y ij when D i = 0 and time= T {3} E(y ij X ij ) = β 0 + β β β β β E(y ij X ij ) = β 0 + β 3 Infererence: Testing Blocks of Coefficients Group time interaction (H 0 : β 4 = β 5 = 0) Time effect (Given β 4 = β 5 = 0, we test H 0 : β 2 = β 3 = 0) Group effect (Given β 4 = β 5 = 0, we test H 0 : β 1 = 0) Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 8 / 18

9 Response Profiles Setting Up These Hypotheses with a Wald Test W 2 = (L β) {LĈov( β)l } 1 (L β) Recall: W 2 χ 2 (r) for r rows of L How to form L? Suppose we had eight parameters (β 0... β 7 ) and wanted to test H 0 : β 5 = β 6 = β 7 = 0 Choose L so that Lβ = 0 expresses the hypothesis. For instance: β 0 β 1 β 2 β 3 β 4 β 5 β 6 β 7 = β 5 β 6 β 7 = Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 9 /

10 Response Profiles Likelihood Ratio Test G 2 = 2(ˆl f ˆl s ) χ 2 (r) for r restrictions Estimate a nested model that imposes your desired restrictions. For example, if your null is H 0 : β 4 = β 5 = 0, then estimate a model that skips the inputs with β 4 and β 5 as coefficients. For this technique, you should use ML, rather than REML. By contrast, the Wald test works with either ML or REML. Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 10 / 18

11 Response Profiles Adjustment for Baseline Response Data with a treatment and a control group often start with a baseline measurement that is pre-treatment for both groups. Consider a study like the National Supported Work Demonstration, when the treatment was rolled out is critical. Are there differences in the baselines? If so, how do we adjust? 1 Keep the baseline and make no assumptions about group differences. 2 Experimental work only: Keep the baseline and assume randomized treatment and control observations have equal averages at baseline. 3 Subtract the baseline response from each post-baseline response, and analyze differences from baseline. 4 Use the baseline value as a predictor of post-baseline responses. Lord s paradox: Different methods of adjustment can yield apparently conflicting results. Why? Some methods tell us whether groups differ in mean change over time. Other methods tell us whether someone in one group is expected to change more or less than a person in another group, given that they have the same baseline response. Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 11 / 18

12 Parametric Curves The Logic of Parametric Curves Does time-dependent variation in average responses follow a pattern? Model that pattern as a polynomial (or perhaps spline) function of time. Relative advantages: Easier interpretation. Added power, particularly at the number of waves increases. Allows for unbalanced designs, even different n i by design. Relative disadvantage: Loses some accuracy relative to response profiles. Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 12 / 18

13 Parametric Curves Linear Trends Over Time Recall: Two features of linear models Nominal Inputs: reference group Interaction Effects: multiply values of a variable Same considerations as response profiles, but a little less messy. Linear Trend Model (G = 2) y ij = β 0 + β 1 D i + β 2 T ij + β 3 D i T ij + ɛ ij Expected Trend for Each Group E(y ij D i = 0) = β 0 + β 2 T ij E(y ij D i = 1) = (β 0 + β 1 ) + (β 2 + β 3 )T ij Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 13 / 18

14 Parametric Curves Upper-Level Polynomials Quadratic Trend Model (G = 2) y ij = β 0 + β 1 D i + β 2 T ij + β 3 T 2 ij + β 4 D i T ij + β 5 D i T 2 ij + ɛ ij Beyond Quadratic Higher-order polynomials are possible. For n waves, the number of time terms cannot exceed n 1. If the order gets particularly high, response profiles may be better. Purpose: Changing Rate of Change (Consider if D i = 0) y ij T = β 2 + 2β 3 T ij y ij (T =1) T = β 2 + 2β 3 y ij (T =2) T = β 2 + 4β 3 Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 14 / 18

15 Parametric Curves Dealing with Collinearity in Time Trend Polynomials If you include T ij, Tij 2, T ij 3,... all in the same model, you can run into some collinearity pretty quickly. Consider: We normally measure time with positive numbers. A quick and reasonably effective fix is to center the time variable such that at the mean, or perhaps in the middle wave, it takes on a value of 0. So create Tij = T ij T. Tij is negative before the middle wave, 0 at the middle wave, and positive thereafter. When creating your polynomial, use Tij when computing exponents. Even-powered terms (e.g., Tij 2 ) will always be non-negative. Odd-powered terms (e.g., Tij 3 ) will have the same sign at each wave as the linear term, albeit with more extreme values. A more thorough fix would be to include orthogonal polynomials of your time trend. See Bock (1975) or Fisher & Yates (1963) for tables. These are computed through Cholesky decomposition. Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 15 / 18

16 Parametric Curves Linear Splines Rather than specify a curves, a line with break points may fit well. Each break point is called a knot. For a knot at time t, add the following term to the linear model: (T ij t ) + = 0 ift ij t (T ij t ) + = T ij t ift ij > t Spline Model with One Knot y ij = β 0 +β 1 D i +β 2 T ij +β 3 (T ij t ) + +β 4 D i T ij +β 5 D i (T ij t ) + +ɛ ij Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 16 / 18

17 Parametric Curves Hypothesis Testing If the response is a linear function of time, a z-ratio can test for a group time interaction. With higher-order polynomials or splines, we need block tests for a group time interaction. (Or even to test a time effect, given no interaction.) We test the same three hypotheses as on Slide 6 and do so using the same types of test. We just don t need to impose as many constraints on in Wald or likelihood ratio tests: Wald: H 0 : Lβ = 0 W 2 = (L β) {LĈov( β)l } 1 (L β) W 2 χ 2 (r) for r rows of L Likelihood Ratio: H 0 : r constraints are true. G 2 = 2(ˆl f ˆl s) χ 2 (r) for r restrictions. This requires the model be (re)estimated with ML, rather than REML. Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 17 / 18

18 Homework Exam Question 2 Download data on electricity consumption in Japan from from Your dependent variable is electricity consumption in GWh in a city in a year (dependent). Your primary input is whether the city received a post-fukushima treatment of a 15% reduction target (any ttdv). Response profiles: Estimate a response profile model where any ttdv is the primary input variable. Estimate with GLS, using REML to estimate the covariance matrix. Plot the mean levels of electricity consumption by combining your coefficient estimates from the linear model. Do a Wald or likelihood ratio test of whether the over-time response profiles are the same for treated and control cities. Parametric curves: Run either a linear, quadratic, or spline model of distance where any ttdv is the primary input variable. Estimate with GLS, using REML to estimate the covariance matrix. Plot the expectations of electricity consumption contingent on treatment and time for each model. Jamie Monogan (UGA) Modeling the Mean Escuela de Invierno 18 / 18