What statistical test should I use to assess rate of photosynthesis in leaf discs?

Question

My biology test is to measure the rate of photosynthesis in leaf discs. It's a simple experiment where I will infiltrate the leaf disc space with sodium-bicarbonate and cause a pressure difference to sink the leaf disc. Then I will shine a light on it causing photosynthesis and time it in minutes for it to rise to the top. EDIT** I will have three leaves of which I cut them all in half. So I will have 6 halves of leaves. Each half of a leaf would have been irradiated so in turn I will have 3 halves which are irradiated, and the other three are left as control. I will then use a cork borer to extract three sample circles from EACH half of a leaf. So I, in turn will have 18 samples. This is the results of ONE sample:

Time/minutes  Has it floated?
0                  No
0.5                No
""
""
7.5                Yes

I have worked out that it will be two tailed since my hypothesis is noting whether there is a change in photosynthesis if irradiated with UV, but other than that, I'm stumped.

Answer 1

Your design appears to have three levels of replication. The first is that you have three leaves, and the second is that each leaf is divided in to paired halves, treated and untreated. The third is that each leaf half is divided into three circles. Presumably each leaf differs from each other, and each half differs from its paired partner half but the difference is less than the difference between leaves, and each circle differs from the other two circles cut from the same leaf half but, again, that difference is less than the difference between leaves.

I assume that you wish to make inferences about the effect of your intervention on leaves. In that case you should consider using a paired analysis so that the influence of the between leaf variability is minimised (or accounted for) by the pairing. The size of the effect of irradiation then needs to be expressed relative to the variability between leaves, not the variability between circles.

The three circles from each leaf half represent what are sometimes called 'technical replicates'. They are not 'proper' experimental units in the way that the leaves are because they have substantial shared variables. They are not independent. A common way to deal with technical replicates is to simply average their values and use that average as the datum. That reduces the variability of the data and increases the power of the analysis. (If you have experience regarding the within leaf variability, you may also use the technical replicates as an informal and internal control for how well the individual experiments went.) If you were to use the technical replicates as full-blown observations you would be guilty of 'pseudo-replication'.

You have one sample of n=3 observations, not 18. Unless the expected effect of the irradiation is large and the variability of the effect of irradiation on leaves is small, the design appears to have fairly low power. However, it sounds like an inexpensive experiment so you could consider expanding it or repeating it.

The exact type of test to use depends on the type of data. If you use the time to float as the observations then you could conceivably just use a paired Student's t-test. However, that assumes continuous values whereas you are proposing to use granular measurements. Can you measure a more exact time to float for each circle? Also, the measurements will not be exactly normally distributed (as they cannot be less than zero), so a transformation may be useful to make the distribution more symmetrical. Which transformation would be best depends on the distribution of the values. Do you have experience of what form that might take? (It cannot be judged on a sample of just 3.)

Student's t-test is quite robust to departures from the assumption of normal distribution, but not so much when the sample is tiny.