# Questions the calculation of Z score and confidence intervals in biology

In biology, especially in some screening assays, people love to calculate a Z score between positive and negative controls. Essentially a distance calculation using the mean and sd to calculate how separated the controls are, hence how strong the signal could be by inferring the range

My question/concern is, shouldn’t we adjust for the actual distribution of the data ? Most the data measured in biology would be from Zero to a positive number, in this case lets say we are measuring light intensity, it ranges from 0 to positive infinity in theory (detection limit, lets say) ,so if the distribution is log normal, then what would happen to the calculated Z score? will we get a smaller/bigger/the same distance? (sorry for my bad drawing)

And the next step is usually report the confidence intervals using the measured controls (usually a couple replicates). Given the log normal distribution, should we adjust that too?

And if so, how to?

• Your title mentions confidence intervals but your actual question doesn't. Is there some background to your question that you may want to add, that brings them in? Commented May 7, 2022 at 10:07
• @ChristianHennig I have made some changes and now showing what I had in my head. It is more like a question if all that matters in the calculation and final interpretation of data. I hope this make sense now :) Commented May 8, 2022 at 7:42
• If you know that data are truly distributed log-normally, you can log-transform your data and then normal theory (as probably used for confidence intervals; I don't know how these are computed for that particular score) will apply. However in a real situation you will not know that, and normal theory may be fine without transformation if data don't look very asymmetric (e.g., when concentrated far away from zero). Commented May 8, 2022 at 14:42

## 2 Answers

My question/concern is, shouldn’t we adjust for the actual distribution of the data?

Yes. For example, this helpful Technical Perspective on Empowering statistical methods for cellular and molecular biologists illustrates different types of data distributions early on, in Figure 1.

What can happen in practice, however, is that there aren't enough data to determine the actual distribution. Or if there are, you might erroneously gauge that the distribution doesn't match the one you had in mind. See, for example, the extensive discussion on whether normality testing is essentially useless.

The Central Limit Theorem and its implications for the distributions of mean-value estimates, as noted by @Christian Hennig, often can allay your concerns substantially. Tests of differences between samples typically depend on normal distributions of sample means rather than of the raw observations themselves.

In your example with an underlying log-normal distribution, the variance among observations of a sample in the original scale could be a function of the mean value. Although that violates assumptions of some hypothesis tests, the practical problems can be minimal. Within any sample the values might be far enough from 0 and closely enough spaced that the normal-distribution assumption isn't too bad within each sample. Welch's t-test, the default 2-sample test in R, can then take any differences of within-sample variances into account.

That said, if your knowledge of the subject matter suggests a particular distribution then it makes sense to analyze data with that distribution in mind. For example, quantitative polymerase chain reaction (qPCR) data are typically expressed in Cq units, the number of reaction cycles needed to pass some fluorescence-intensity threshold. Those Cq values are logarithmically related (inversely) to the amount of PCR target, with each halving of the target amount leading to a 1 unit increase in Cq.

I routinely do calculations in the Cq scale rather than the target-amount scale, as the data in Cq scale tend to meet the assumptions of regression models better. Residuals in that scale, rather than in the target-amount scale, are more likely to meet the assumption of constant variance across the range of observations. If I need to translate to the target-amount scale, I calculate confidence intervals in the more symmetric Cq scale first and then transform to target amounts.

Generalized linear models (GLM) can allow for explicit handling of specific data distributions. For example, with Poisson-distributed count data (also seen frequently in biology), a Poisson GLM can incorporate the known equality of the mean value and the variance. Logistic regression for binary outcomes incorporates the variance that arises from binomial sampling. With your well-founded concern over how data distributions matter, look into how such models handle this issue and how they also allow for more general relationships between predictor variables and outcome measures.

Mean and sd have an interpretation that isn't exclusive to the normal distribution, so I don't think such a Z score should be seen as requiring normality (normality may be required for certain distributional statements for example used for computing confidence intervals and the like, but those can hold approximately also for other distributions due to the Central Limit Theorem).

However it is true that mean and sd are problematic for summarising a distribution's location and variation for clearly skew distributions or in presence of outliers, so in such cases such scores may be inappropriate. Ultimately it depends on the exact situation and the use made of and interpretation given to these scores.

For example, certain data distributions may be more symmetric and show fewer problems with outliers after a log transformation, but even then in certain applications interpretation is relevant in terms of the original values rather than in terms of logs.

• thank you and sorry I was posting on my phone and got distracted so I forgot the write the CV part. Like you said, I think mean and sd is just a description on the distribution, normal or not. So Z score is fine to say how far away the means are between 2 distributions. But the CVs are the ones I worry, if I should adjust for the log normal like distribution. In biology, I feel a lot the times we see "ourliers" making the tails fat. Do you reckon if I should just do a log transformation before I calculate CV? Commented May 7, 2022 at 12:54
• (a) Don't use abbreviations that you don't explain. It seems you use CV for confidence interval, but that's for sure not a good and clear abbreviation. (b) Without knowing your context and data I have no idea whether a log transformation improves matters or not. Commented May 7, 2022 at 15:04
• Also it is good practice to add additional information to your question rather than mentioning it only in comments (that some people may not read). Commented May 7, 2022 at 15:05
• Thank you. I will arrive home today after my trip (and my computer), then I can update this question with more details. Thank you for the tips Commented May 8, 2022 at 2:16