We are counting the numbers of cells (oligodendrocytes) that develop in mutant mice versus littermate controls, and wish to know if the numbers are different. Typically we can get 2 mutants and 2 controls in each litter (of ~8 pups; mutants are compound). The ratio mutant/control within each litter is consistently ~0.75 (ie 25% reduction in mutant). Because of variation among litters (developmental age, which is difficult to determine precisely; slightly different background genotypes; different fixation and immunolabelling conditions) we do not find a statistically significant effect (t-test) when we calculate the ratio as (M1+M2+M3 ...+Mn)/(C1+C2+C3..+Cn), where Mn is the mean of the cell numbers in the mutant in a given litter (n litters total) and Cn is the mean of the control values in the same litter). However, we do have a significant effect if we calculate the ratio within each litter separately and then take the mean ratio i.e. (M1/C1+M2/C2+M3/C3..+Mn/Cn)/n My questions are: 1) how would one describe these two methods of calculating the mean ratio? i.e. is there a recognized term for each approach? 2) What is the appropriate statistical method to use in each case (e.g. is sample size "n" the same in both approaches, or should we use the number of separate litters, rather than mice, in the latter method?
1 Answers
Sometimes when dealing with ratios it helps to log-transform your data. For example, defining the cell counts from the $i^{th}$ mouse as $y_i$, you could model the cell counts as follows:
$$y_i = \beta_0 + \beta_1 (\text{mouse}_i \text{ is mutant}) + \beta_2 (\text{mouse}_i \text{ is in litter 1}) + \beta_3 (\text{mouse}_i \text{ is in litter 2}) + \dots + \beta_8 (\text{mouse}_i \text{ is in litter 7}) + e_i$$
The $\beta$ 's can be estimated using ANOVA with mutation state and litter membership as the sources of variation. Now consider two mice 'typical' mice from litter 1 where one of which is a mutant and the other is not. According to the model, ratio of their cell counts is
$$\frac{\exp(y_m)}{\exp(y_c)} = \frac{\exp(\beta_0 + \beta_1 + \beta_2)}{\exp(\beta_0 + \beta_2)} = \exp({\beta_1})$$
Therefore, by exponentiating $\beta_1$, you can estimate the 'average' ratio between mutant cell counts and control cell counts. Also, you can show that this estimated ratio is statistically different from 1 by verifying that the confidence interval for $\beta_1$ does not contain 0.
[As an aside: I don't think the paired-data tag is appropriate. I associate the term 'paired data' with paired measurements taken from the same measurement unit. For example, suppose you could somehow induce the mutation after the mice were born. You could count the number of cells before inducing the mutation and then again afterwards. Each 'before' measurement is paired with an 'after' measurement since they were taken from the same mouse. In your case, there is no pairing between measurements. Suppose Litter 1 has four mice: two controls (C11 and C12) and two mutants (M11 and M12). There's no justification for pairing the cell count from C11 with the count from M11 as opposed to the count from M12.]
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