Harold, I think we use slightly different notation (I like to use variance parameters rather than covariance matrices). Let me try to write it in model form:
Data points y_i, i=1,...,800 800 x 84 matrix of predictors, X: for columns j=1,...,82, X_{i,j} is the amount of food j consumed by person i. X_{i,83} is an indicator (1 if male, 0 if female), and X_{i,84} is the age of person i. Data-level model: Pr (y_i=1) = inverse.logit (X_i*beta), for i=1,...,800, with independent outcomes. beta is a (column) vector of length 84. Group-level model: for j=1,...,82: beta_j ~ Normal (gamma_0 + gamma_1 * u_j, sigma^2_{beta}). u is a vector of length 82, where u_j = folate concentration in food j gamma_0 and gamma_1 are scalar coefficients (for the group-level model), and sigma_{beta} is the sd of the group-level errors. It would be hopeless to estimate all the betas using maximum likelihood: that's 800 data points and 84 predictors, the results will just be too noisy. But it should be ok using the 2-level model above. The question is: can I fit in lmer()? Thanks again. Andrew Doran, Harold wrote: > So, in the hierarchical notation, does the model look like this (for > the linear predictor): > > DV = constant + food_1(B_1) + food_2(B_2) + ... + food_82(B_82) + > sex(B_83) + age(B_84) > food_1 = gamma_00 + gamma_01(folic) + r_01 > food_2 = gamma_10 + gamma_11(folic) + r_02 > ... > food_82 = gamma_20 + gamma_21(folic) + r_82 > > where r_qq ~ N(0, Psi) and Psi is an 82-dimensional covariance matrix. > > I usually need to see this in model form as it helps me translate this > into lmer syntax if it can be estimated. From what I see, this would > be estimating 82(82+1)/2 = 3403 parameters in the covariance matrix. > > What I'm stuck on is below you say it would be hopeless to estimate > the 82 predictors using ML. But, if I understand the model correctly, > the multilevel regression still resolves the predictors (fixed > effects) using ML once estimates of the variances are obtained. So, I > feel I might still be missing something. > > > > -----Original Message----- > From: Andrew Gelman [mailto:[EMAIL PROTECTED] > Sent: Sun 5/21/2006 7:35 PM > To: Doran, Harold > Cc: r-help@stat.math.ethz.ch; [EMAIL PROTECTED] > Subject: Re: [R] Can lmer() fit a multilevel model embedded in > a regression? > > Harold, > > I get confused by the terms "fixed" and "random". Our first-level model > (in the simplified version we're discussing here) has 800 data points > (the persons in the study) and 84 predictors: sex, age, and 82 > coefficients for foods. The second-level model has 82 data points (the > foods) and two predictors: a constant term and folic acid concentration. > > It would be hopeless to estimate the 82 food coefficients via maximum > likelihood, so the idea is to do a multilevel model, with a regression > of these coefficients on the constant term and folic acid. The > group-level model has a residual variance. If the group-level residual > variance is 0, it's equivalent to ignoring food, and just using total > folic acid as an individual predictor. If the group-level residual > variance is infinity, it's equivalent to estimating the original > regression (with 84 predictors) using least squares. > > The difficulty is that the foods aren't "groups" in the usual sense, > since persons are not nested within foods; rather, each person eats many > foods, and this is reflected in the X matrix. > > Andrew > > Doran, Harold wrote: > > > OK, I'm piecing this together a bit, sorry I'm not familiar with the > > article you cite. Let me try and fully understand the issue if you > > don't mind. Are you estimating each of the 82 foods as fixed effects? > > If so, in the example below this implies 84 total fixed effects (1 for > > each food type in the X matrix and then sex and age). > > > > I'm assuming that food type is nested within one of the 82 folic acid > > concentrations and then folic acid is treated as a random effect. > > > > Is this accurate? > > > > > > -----Original Message----- > > From: Andrew Gelman [mailto:[EMAIL PROTECTED] > > Sent: Sun 5/21/2006 9:17 AM > > To: Doran, Harold > > Cc: r-help@stat.math.ethz.ch; [EMAIL PROTECTED] > > Subject: Re: [R] Can lmer() fit a multilevel model embedded in > > a regression? > > > > Harold, > > > > I'm confused now. Just for concretness, suppose we have 800 people, 82 > > food items, and one predictor ("folic", the folic acid concentration) at > > the food-item level. Then DV will be a vector of length 800, foods is > > an 800 x 82 matrix, sex is a vector of length 800, age is a vector of > > length 800, and folic is a vector of length 82. The vector of folic > > acid concentrations in individual diets is then just foods%*%folic, > > which I can call folic_indiv. > > > > How would I fit the model in lmer(), then? There's some bit of > > understading that I'm still missing. > > > > Thanks. > > Andrew > > > > > > Doran, Harold wrote: > > > > > Prof Gelman: > > > > > > I believe the answer is yes. It sounds as though persons are partially > > > crossed within food items? > > > > > > Assuming a logit link, the syntax might follow along the lines of > > > > > > fm1 <- lmer(DV ~ foods + sex + age + (1|food_item), data, family = > > > binomial(link='logit'), method = "Laplace", control = list(usePQL= > > > FALSE) ) > > > > > > Maybe this gets you partly there. > > > > > > Harold > > > > > > > > > > > > -----Original Message----- > > > From: [EMAIL PROTECTED] on behalf of Andrew Gelman > > > Sent: Sat 5/20/2006 5:49 AM > > > To: r-help@stat.math.ethz.ch > > > Cc: [EMAIL PROTECTED] > > > Subject: [R] Can lmer() fit a multilevel model embedded in a > > > regression? > > > > > > I would like to fit a hierarchical regression model from Witte et al. > > > (1994; see reference below). It's a logistic regression of a health > > > outcome on quntities of food intake; the linear predictor has the > form, > > > X*beta + W*gamma, > > > where X is a matrix of consumption of 82 foods (i.e., the rows of X > > > represent people in the study, the columns represent different foods, > > > and X_ij is the amount of food j eaten by person i); and W is a matrix > > > of some other predictors (sex, age, ...). > > > > > > The second stage of the model is a regression of X on some food-level > > > predictors. > > > > > > Is it possible to fit this model in (the current version of) lmer()? > > > The challenge is that the persons are _not_ nested within food > items, so > > > it is not a simple multilevel structure. > > > > > > We're planning to write a Gibbs sampler and fit the model > directly, but > > > it would be convenient to be able to flt in lmer() as well to check. > > > > > > Andrew > > > > > > --- > > > > > > Reference: > > > > > > Witte, J. S., Greenland, S., Hale, R. W., and Bird, C. L. (1994). > > > Hierarchical regression analysis applied to a > > > study of multiple dietary exposures and breast cancer. > Epidemiology 5, > > > 612-621. > > > > > > -- > > > Andrew Gelman > > > Professor, Department of Statistics > > > Professor, Department of Political Science > > > [EMAIL PROTECTED] > > > www.stat.columbia.edu/~gelman > > > > > > Statistics department office: > > > Social Work Bldg (Amsterdam Ave at 122 St), Room 1016 > > > 212-851-2142 > > > Political Science department office: > > > International Affairs Bldg (Amsterdam Ave at 118 St), Room 731 > > > 212-854-7075 > > > > > > Mailing address: > > > 1255 Amsterdam Ave, Room 1016 > > > Columbia University > > > New York, NY 10027-5904 > > > 212-851-2142 > > > (fax) 212-851-2164 > > > > > > ______________________________________________ > > > R-help@stat.math.ethz.ch mailing list > > > https://stat.ethz.ch/mailman/listinfo/r-help > > > PLEASE do read the posting guide! > > > http://www.R-project.org/posting-guide.html > > > > > > > > > > -- > > Andrew Gelman > > Professor, Department of Statistics > > Professor, Department of Political Science > > [EMAIL PROTECTED] > > www.stat.columbia.edu/~gelman > > > > Statistics department office: > > Social Work Bldg (Amsterdam Ave at 122 St), Room 1016 > > 212-851-2142 > > Political Science department office: > > International Affairs Bldg (Amsterdam Ave at 118 St), Room 731 > > 212-854-7075 > > > > Mailing address: > > 1255 Amsterdam Ave, Room 1016 > > Columbia University > > New York, NY 10027-5904 > > 212-851-2142 > > (fax) 212-851-2164 > > > > > > > > -- > Andrew Gelman > Professor, Department of Statistics > Professor, Department of Political Science > [EMAIL PROTECTED] > www.stat.columbia.edu/~gelman > > Statistics department office: > Social Work Bldg (Amsterdam Ave at 122 St), Room 1016 > 212-851-2142 > Political Science department office: > International Affairs Bldg (Amsterdam Ave at 118 St), Room 731 > 212-854-7075 > > Mailing address: > 1255 Amsterdam Ave, Room 1016 > Columbia University > New York, NY 10027-5904 > 212-851-2142 > (fax) 212-851-2164 > > > -- Andrew Gelman Professor, Department of Statistics Professor, Department of Political Science [EMAIL PROTECTED] www.stat.columbia.edu/~gelman Statistics department office: Social Work Bldg (Amsterdam Ave at 122 St), Room 1016 212-851-2142 Political Science department office: International Affairs Bldg (Amsterdam Ave at 118 St), Room 731 212-854-7075 Mailing address: 1255 Amsterdam Ave, Room 1016 Columbia University New York, NY 10027-5904 212-851-2142 (fax) 212-851-2164 ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html