Solved – How to interpret slope parameter estimates for linear models in R

interpretationmixed modelparameterizationr

I wish to analyse a simple lab experiment. I have 8 fish. Four are fed on diet A, and four on diet B. I measure their Nitrogen (N) over 5 time periods (so 5 repeated measures per fish). I wish to know 3 pieces of information:

1) On diet A, does N change with time (i.e. a true slope different from zero)?

2) On diet B, does N change with time (i.e. a true slope different from zero)?

3) Do slopes of A and B differ from one another?

Thus I have run a simple mixed model in R, of the form:

Nitrogen ~ Time * Diet + (1|replicate)

Replicate is used as a random effect to control for repeated mesaures per individual fish.

I have compared this model to a simpler model without an interaction term, using analysis of deviance:

Nitrogen ~ Time + Diet + (1|replicate)

The model with the interaction explained signifcantly more variance and thus the better model.

Based on this interaction model, I have the following output table with the parameter estimates, and I wondered if I can use this table to answer my three questions? (I have calculated confidence intervals for these parameter estimates to determine if effects are real).

Fixed effects:                              
             Estimate  Std. Error   t value   +95% CI     -95% CI
(Intercept)  15.8624     0.2332    68.0300    16.3194     15.4053
time          0.0069     0.0009     7.7600     0.0086      0.0051
dietB        -0.1948     0.3298    -0.5900     0.4515     -0.8411
time:dietB   -0.0066     0.0013    -5.2800    -0.0042     -0.0091

I understand the "Intercept" is value of N when time = 0 for diet A (R software works alphabetically so dietA before dietB), while "dietB" + Intercept is intercept for dietB when time = 0.

However, to answer my 3 questions I need to better understand how to interpret the slope information and how to report it. My current understanding is:

"time" gives me a real slope value (0.0069) to report for diet A, and having calculated confidence intervals, I can see they do not cross zero in this instance meaning diet A is a true slope (i.e. different from a slope of zero). So this is Question 1 answered, I hope?

"time:dietB" tells me that slope of dietB is 0.0069("time") + -0.0066, which gives me a slope value to report for diet B (0.0069 + – 0.0066 = 0.0003). As I currently understand it, the Std. Error for "time:dietB" is associated with this comparative slope value (-0.0066), and not on slope B's real value (0.0003), and consequently, confidence intervals also refer to this comparative value (-0.0066), and thus as confidence intervals do not pass through zero, this means that slope B is different from slope A. So this means Question 3 is answered, I hope?

If this is the case, then how do I answer my Question 2 – that slope B is different from a slope of zero? Can I get confidence intervals to answer this question, and/or do I need them?! Or is this information already contained within this summary table? Or instead, do I need to conduct a subsequent / entirely different analysis?

Best Answer

I think it's always difficult to interpret the size of interaction effects, but it can be done in a different way to make it easier.

If you create two new variables where the first is 0 for diet B and equal to time for diet A, and the other variable is the oppoosite, you assess the effect of the slope for each diet more directly:

TimeA <- ifelse(Diet=="A", Time, 0)
TimeB <- ifelse(Diet=="B", Time, 0)

Nitrogen ~ Diet + TimeA + TimeB + (1|replicate)

You will now get separate estimates for the slope for time A and time B, with standard errors so that you can calculate confidence intervals to report. You will see that the estimates for Diet are the same, and the estimate for TimeA is the same as the estimate for Time in your model. The difference is that in this model, you will get one estimate for the slope of time for Diet B, and it will probably be non-significant in your model.

Note that this assumes that the baseline for Time is 0. If not, the results will be more difficult to interpret.

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how can I know which ones are significantly different from zero, and how would I report this?

To obtain the slope estimate and its statistical significance for each level of a factor, you can perform the following tests:

testInteractions(test1, custom=list(func.group=c(1,0,0)), slope="sr", adjustment="none")
testInteractions(test1, custom=list(func.group=c(0,1,0)), slope="sr", adjustment="none")
testInteractions(test1, custom=list(func.group=c(0,0,1)), slope="sr", adjustment="none")

You may report the results of the individual slopes the same way as the pairwise comparisons of slopes (e.g., tables).

Solved – On the utility of the intercept-slope correlation in multilevel models

I have emailed several scholars (almost 30 persons) several weeks ago. Few of them sent their mail (always collective emails). Eugene Demidenko was the first to answer :

cov/sqrt(var1*var2) is always within [-1,1] regardless of the interpretation: it may be estimates of intercept and slope, two slopes, etc. The fact that -1<=cov/sqrt(var1*var2)<=1 follows from the Cauchy inequality and it is always true. Thus I dismiss the Snijders & Bosker statement. Maybe some other piece of information is missing?

This was followed by an email from Thomas Snijders :

The information that is missing is what was actually written about this on page 122, 123, 124, 129 of Snijders & Bosker (2nd edition 2012). This is not about two competing claims of which no more than one can be true, it is about two different interpretations.

On p. 123 a quadratic variance function is introduced, \sigma_0^2 + 2 \sigma_{01} * x + \sigma_1^2 * x^2 and the following remark is made: "This formula can be used without the interpretation that \sigma_0^2 and \sigma_1^2 are variances and \sigma_{01} a covariance; these parameters might be any numbers. The formula only implies that the residual variance is a quadratic function of x.

Let me quote a full paragraph of p. 129, about a quadratic variance function at level two; note that ONE MIGHT INTERPRET that \tau_0^2 and \tau_1^2 are the level-two variances of the random intercept and random slope, and \tau_{01} is their covariance, but this is explicitly put behind the horizon:

"The parameters \tau_0^2, \tau_1^2, and \tau_{01} are, as in the preceding section, not to be interpreted themselves as variances and a corresponding covariance. The interpretation is by means of the variance function (8.7) [note t.s.: in the book this is mistakenly reported as 8.8]. Therefore it is not required that \tau_{01}^2 <= \tau_0^2 * \tau_1^2. To put it another way, 'correlations' defined formally by \tau_{01}/(\tau_0 * \tau_1) may be larger than 1 or smaller than -1, even infinite, because the idea of a correlation does not make sense here. An example of this is provided by the linear variance function for which \tau_1^2 = 0 and only the parameters \tau_0^2 and \tau_{01} are used."

The variance function is a quadratic function of x (the variable "with the random slope"), and the variance of the outcome is this plus the level-1 variance. As long as this is positive for all x, the modelled variance is positive. (An extra requirement is that the corresponding covariance matrix is positive definite.)

Some further background of this is the existence of differences in parameter estimation algorithms in software. In some multilevel (random effects) software, the requirement is made that the covariance matrices of the random effects are positive semi-definite on all levels. In other software, the requirement is made only that the resulting estimated covariance matrix for the observed data is positive semi-definite. This implies that the idea of random coefficients of latent variables is relinquished, and the model specifies a certain covariance structure for the observed data; no more, no less; in that case the cited interpretation of Joop Hox does not apply. Note that Harvey Goldstein already long ago used linear variance functions at level one, represented by a zero slope variance and nonzero slope-intercept correlation at level one; this was and is called "complex variation"; see, e.g., http://www.bristol.ac.uk/media-library/sites/cmm/migrated/documents/modelling-complex-variation.pdf

And then, Joop Hox replied :

In the software MLwiN it is actually possible to estimate a covariance term and at the same time constrain one of the variances to zero, which would make the "correlation" infinite. And yes, some software will allow estimates such as negative variances (SEM software usually allows this). So my statements were not completely accurate. I refered to "normal" unstructured random structures. Let me add that if you rescale the variable with the random slope to have a different zero-point, the variances and covariances generally change. So the correlation is only interpretable if the predictor variable has a fixed zero-point, i.e. is measured on a ratio scale. This applies to growth curve models, where the correlation between initial status and rate of growth is sometimes interpreted. In that case the value zero should be the 'real' time point where the process starts.

And he sent another mail :

Anyway, I think Tom's explanation below fits the style of the Snijders/Bosker collaboration better than my more informal style. I would add to page 90 a footnote stating something like "Note that the parameter values in the random part are estimates. Interpreting the standardized covariances as ordinary correlations assumes that there are no constraints on the variances and that the software does not allow negative estimates. If the random part is unstructured the interpretation as ordinary (co)variances is generally tenable.".

Note that I wrote about the correlation interpretation in the longitudinal chapter. In growth curve modeling it is very tempting to interpret this correlation as a substantive result, and that is dangerous because the value depends on the "metric of time". If you are interested in that I recommend to go to Lesa Hoffman's website (http://www.lesahoffman.com/).

So I think in my situation, where I've specified an unstructured covariance for the random effects, I should interpret the intercept-slope correlation as an ordinary correlation.

Best Answer

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