longitudinal with 2 time points
John, That you are asking this question indicates that either you have yet to read the article or that you need to re-read it, as you have not comprehended the content. The beta coefficient for treatment IS the difference in mean glucose change between baseline and 4 weeks **attributable to treatment**, after adjusting for any baseline differences in glucose between the two groups. That is also presuming that there is no interaction at baseline. For example, let's say that the beta for treatment is -20. Then, at 4 weeks, given the same baseline glucose level, we would predict that, on average, the treatment group will have a glucose level 20 mg/dl less than the control group. In the absence of an interaction, we would estimate the same average treatment difference at 4 weeks of 20 mg/dl whether the baseline glucose was 300 mg/dl or 100 mg/dl. However, given regression to the mean, we might reasonably expect the patient with a 300 mg/dl baseline level to have a greater mean reduction at 4 weeks as compared to the patient with a 100 mg/dl baseline level. We might also expect a patient with a glucose level at the low end of the baseline range (eg. 50 mg/dl) to experience an average increase in glucose level at 4 weeks, presuming that your inclusion/exclusion criteria permitted patients with below normal glucose levels. But the difference will still be, on average, 20 mg/dl between the two treatment groups. So the patient with a 300 mg/dl baseline level might have an average reduction to 200 mg/dl at 4 weeks on the control treatment, whereas the same patient on the active treatment would have an average reduction to 180 mg/dl (a difference of -20). The patient with a 100 mg/dl baseline level might have an average reduction to 90 mg/dl at 4 weeks on the control treatment, whereas the same patient on the active treatment would have an average reduction to 70 mg/dl (again, a difference of -20). The patient with a 50 mg/dl baseline level might have an average increase to 90 mg/dl at 4 weeks on the control treatment, whereas the same patient on the active treatment would have an average increase to 70 mg/dl (yet again, a difference of -20). So your conclusion would be that on average, between baseline and 4 weeks, glucose levels were reduced by 20 mg/dl more in the active treatment group relative to control. This difference is the vertical separation in the two parallel fitted regression lines as shown in the figure in the paper. So the method is answering exactly the question the investigator is asking. Marc
On Aug 13, 2010, at 1:02 AM, array chip wrote:
Marc, Thanks for sharing your insights. Let's take this model as an example: lm(wk4.glucose ~ baseline.glucose + treatment + gender + age) Because the investigator is interested in knowing whether the CHANGE of glucose in week 4 from baseline is different between treatment and control, Is it still legitimate to ask whether and HOW can we test this hypothesis? I think the coefficient of the treatment factor is only testing whether the week 4 glucose level is different between treatment and control, but not testing whether the CHANGE of week 4 glucose level with respect to baseline is different between treatment and control. Thanks again for your suggestion. Yi ----- Original Message ---- From: Marc Schwartz <marc_schwartz at me.com> To: array chip <arrayprofile at yahoo.com> Cc: Charles E. (Ted) Wright <cewright at uci.edu>; John Maindonald <john.maindonald at anu.edu.au>; r-sig-mixed-models at r-project.org Sent: Thu, August 12, 2010 6:02:29 AM Subject: Re: [R-sig-ME] longitudinal with 2 time points Hi John, If you read that article, you will see that your use of delta.y as the dependent variable does not make sense. Thus, I would re-express your model 5 as: lm(wk4.glucose ~ baseline.glucose + treatment + gender + age) and as noted, check for the interaction between baseline glucose and treatment: lm(wk4.glucose ~ baseline.glucose * treatment + gender + age) You might also want to consider using a spline function on age, presuming that age is hopefully measured as a continuous variable (eg. not ordinal groups). Since the ANCOVA based approach described in the paper is essentially an OLS linear regression, you can of course include the additional covariates for adjustment. If the interaction term p value is >0.1 (a common threshold), you can remove it and the beta coefficient and its CIs for the treatment factor is your estimated treatment effect relative to your control. For the presentation of the results, besides the obvious tabular summaries and the scatter/regression lines plot, include a series of plots showing selected baseline values and the treatment versus control predicted follow up values and CIs for the same baseline value in each plot. This visually shows the common estimated treatment effect for each baseline value, which will also tend to reveal regression to the mean. This presentation is especially helpful if the interaction term is retained, which therefore shows how the treatment effect varies and will reverse, over the range of the baseline values. You can select a series of clinically relevant values over the range of the observed baseline values, and/or by default, select a five number plus mean series over the observed baseline values. I don't see a role for a mixed effects model here, given that this is a pretty straightforward "change from baseline" type design, but there are many here with greater expertise than I. If this was a cross-over design, you have multiple measures of glucose for each patient at each time point, more than two time points, or a multi-center study, then a mixed effects model would make more sense to me. HTH, Marc On Aug 12, 2010, at 12:39 AM, array chip wrote:
Hi Marc, Thanks for the reference. I will definitely read it. Please see my reponse to John's reply. Your model is another model I should add to the 5 models I proposed in that email. What's your overall thoughts on these different
models?
Thank you for sharing. John ----- Original Message ---- From: Marc Schwartz <marc_schwartz at me.com> To: Charles E. (Ted) Wright <cewright at uci.edu>; array chip <arrayprofile at yahoo.com> Cc: John Maindonald <john.maindonald at anu.edu.au>; r-sig-mixed-models at r-project.org Sent: Wed, August 11, 2010 6:20:13 AM Subject: Re: [R-sig-ME] longitudinal with 2 time points Hi, I'll throw in a reference that covers some of these issues: Statistics Notes Analysing controlled trials with baseline and follow up measurements Vickers and Altman BMJ. 2001 November 10; 323(7321): 1123?1124. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1121605/ The basic model specification would of course be: lm(4Wks ~ Baseline + Group) You will also want to test for an interaction between the baseline score and your grouping factor, in case the observed group (eg. treatment) effect is dependent upon the value of the baseline measurement. In this case, unlike in the above paper, you of course end up with crossing fitted regression lines, rather than parallel lines. HTH, Marc Schwartz On Aug 11, 2010, at 7:34 AM, Charles E. (Ted) Wright wrote:
Keep in mind that running an ANOVA on the difference is not the same thing as
using the baseline data as a covariate in an ANOVA on the Week 4 data. Essentially the ANOVA on the differences is like the ANCOVA with the slope constrained to be 1. Ted Wright On Wed, 11 Aug 2010, John Maindonald wrote:
All these are possibilities, except maybe making baseline measurement a random factor. This would make sense only if data divide into groups, and you want the baseline effect to vary randomly from group to group. That may limit your ability to estimate parameters that are of interest. In most circumstances that I am familiar with, it makes better sense to treat baseline effect as fixed. John. On 11/08/2010, at 8:11 AM, array chip wrote:
Hi, I am wondering if it is still meaningful to run a mixed model if a longitudinal dataset has only 2 time points (baseline and week 4)? Would it
be
more appropriate to simply take the difference between the 2 time points and
run ANOVA (ANCOVA) on the difference? what about still running mixed model on
the
difference of the 2 time points, but adding baseline measurement as a
random
factor? Thanks for sharing your thoughts. John