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I would like some help with the following problem.

I have 40 subjects. On each subject I take a measurement at 25 body sites. The measurement is a continuous variable that varies between 0 and 1 and appears to be normally distributed. I want do a test to see if body site statistically significantly affects the measurement. I also have demographic data on the subjects but haven't tried to analyze that yet.

I think the correct thing to do would be to use repeated measures ANOVA or MANOVA to do the analysis because the data is paired. The problem is that I have missing data and my basic understanding is that I would have to exclude any subject that has data missing for even one body site. If I did this, I would only have 6 subjects left. If I excluded any body site with data missing then I would only have 8 body sites left.

So my thought was that I could use regular ANOVA but just lose some power to see a difference. However, this seems like it violates the assumption of independence for ANOVA. So my main questions are as follows:

  1. Does using Regular ANOVA on matched data violate the independence assumption, and if so is the only consequence a decreased likelihood of rejecting the null hypothesis or is there more to it than that? Also does anyone know of a reference I could site regarding this?

  2. Does anyone have any suggestions for a better way to analyze my data? By the way I usually use JMP or SPSS for analysis.

Thanks!

Billy

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  • $\begingroup$ This does not directly pertain to the violation of independence, but have you considered multiple imputation to deal with your missing values? $\endgroup$ – Christopher Aden Jan 15 '11 at 2:57
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    $\begingroup$ Why is there missing data? Is the data missing completely at random, or can the reason for a given observation being missing be explained by the non-missing data or the demographic data? The best method for handling missing data depends critically on the reason for missingness. $\endgroup$ – onestop Jan 15 '11 at 8:42
  • $\begingroup$ I shine light on the skin and measure the properties of the light returning. The property I am analyzing is independent of the the mean intensity of the light unless it is below a certain intensity. Thus the device rejects any data with a mean intensity below the cutoff. Mean intensity is determined by the exposure time, set manually by a tech. Sometimes he chooses poorly and the mean intensity is too low. For the most part the missing values are randomly distributed throughout except two body sites need longer exposure times and he is more likely to underestimate the needed exposure time. $\endgroup$ – Billy Jan 16 '11 at 1:41
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Billy,

From the comment about the data being rejected if the light is below a certain intensity, does this correspond to why you have missing records? Because if you do, then you do not have total "missingness", but rather "censoring" because you know that the response is below a certain threshold. This does have an impact because the likelihood function becomes a product of the density functions for those observed value, and a product of the cumulative density functions for those values which were "rejected". I would say that this is a case of the data being "Not Missing At Random" (NMAR). Basically this means the cause of the missingness is related to the actual value that is missing, which in this case, the value is below a certain threshold.

One thing I find interesting about NMAR is it often comes across as a "bad" thing. My view is that it is actually a good thing (from a statistical point of view) because you haven't completely lost the record, it is still giving you some information. The only bad thing is that the "standard" mathematics usually don't work as elegantly, and standard software can't be used as easily. You just need to work harder to extract the information.

Another thing you make reference to is the "mean intensity" of the light. It seems odd to talk of "mean" when referring to a single measurement. Or is this the mean of the light over the area of the spot?

Survival analysis with censored data may be a useful place to start, since this is a similar problem mathematically. It may be a bit of a hurdle to "translate" the survival application into a relevant one for your analysis.

My advice would be to start simple, and build complexity as you go. It seems as though you have already done this a bit (by excluding the demographic data initially).

One way to do this is to do a simple one-way ANOVA only using complete cases with the classification over the 25 sites. one-way unbalanced ANOVA is quite easy to implement (but not as a regression! lack of balance makes ANOVA different from its standard regression representation!). It resembles the two sample t-test with different sample sizes. Another option is to analyse the data 2 body sites at a time using complete "paired" cases. This would give you a kind of "ordering" of the body types.

This analysis is all of the "exploratory" type, because it essentially "throws away" part of the information in exchange for a more simple view of the data.

It does seem as though this would be quite an involved analysis to take all of the information into account (particularly the censoring). The multi-level approach suggested by @BR would be a useful approximation to this (by "throwing away" the threshold information). The multiple imputation approach suggested by Christopher Aden is another way to go, where you get the advantage of the easier to understand ANOVA, while properly taking account of the uncertainty due to missing values.

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  • $\begingroup$ Thanks for the thoughtful input. In terms of the words "mean intensity," you are correct that I mean the mean of the light across the area of a spot (not of multiple subjects). In terms of the censoring of the data, I think I didn't explain it clearly. My variable of interest isn't actually the intensity. It is a different variable called contrast. But when the intensity is too low, I am unable to reliably measure the contrast( I know this from previous experiments on objects with known properties). Thus the values of the contrast, when the intensity is too low, are censored. $\endgroup$ – Billy Jan 16 '11 at 23:09
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You should consider using mixed-effect / multi-level models. The techniques used to fit these models work fine with unbalanced design, which is how it will interpret your missing data. As long as the data is missing at random, this is a reasonable way to proceed. SPSS is able to fit linear mixed-effect models.

Mixed-effect models also allow continuous covariates, so you could add, e.g., age as a person-level predictor.

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