# Can I use z-scores to compare data measured in different units?

I am trying to compare the accuracy of two different methods of estimating biomass. This data is in the form of percentage cover (and is arcsine transformed). I also have the real biomass values, measured in grams, so three different measurements of the same thing, one of which (for the purpose of the study) I'm assuming provides the correct values, the other two are of unknown accuracy. What's the best way to approach this?

I've tried two separate correlation analyses, and then used Fisher's r-to-z transformation to compare the correlation coefficients.

I was wondering if it's possible to calculate an 'accuracy' value by standardising all the scores, then taking the absolute value of the difference between my (standardised) % cover data points and the (standardised) biomass value in grams. This would be useful because then I could see whether the accuracy changes over time, for example. But does it make sense?

I'm a statistics novice and could really do with some tips! I'm using SPSS.

If I understand this correctly, you have

$B$. Measured values of biomass in grams, which you regard as your best measurement.

$C_1$ and $C_2$. Two measures of percent cover, which you regard as proxies for biomass.

The main question is surely the relationships between $B$ on the one hand and $C_1$ and $C_2$ on the other. As $B$ and $C_1$, $C_2$ are in quite different units of measurement, you can only assess accuracy by establishing what functions best approximate $B$ as predicted from $C_1$ and $C_2$ respectively.

Calculating a correlation is pertinent to the extent that the relationships are approximately linear, and my guess would be to doubt that, especially as $B$ is unbounded (no mathematical limit on its upper values) whereas the $C$s are bounded. I'd expect some kind of nonlinear relation here.

Transforming percents or correlations are secondary details; it may be that arcsine transformations (by which you may well mean "arcsine of square root" as that is a common transformation here) help with nonlinearity, or it may not be. Nor is it clear that any standardisation will help at all: standardisation of each variable separately makes sense if there are linear relationships of the form $\hat B = a + bC$, but if so you are better off working directly with those relationships.

To give good advice we need to see plots of $B$ vs $C_1$ and $B$ vs $C_2$; there is, in my view, no correct method for this problem that can be given in abstraction. Listings of the data would be most helpful if they can be provided easily.

• Thanks for this. Yes, you've correctly understood my data and the question I'm trying to answer. I'm posting scatter graphs for the relationships between B (weight) and the untransformed C1 and C2. As far as I can tell there is a linear relationship (although there seems to be heteroscedasity?) Jan 23 '15 at 12:36
• I'd fit power functions here of the form $B = aC^b$ which have the right limiting behaviour $C = 0 \Rightarrow B = 0$. What you have shown suggests that the bounds on cover don't bite as it is not more than $0.3$. Jan 23 '15 at 12:51
• Sorry, can you explain that a bit more fully? I don't understand what you're suggesting- how do i find the appropriate a and b values? Jan 23 '15 at 12:55
• Usually people take logarithms and fit a regression, as $\log B = \log a + b \log C$. As I guess you're a biologist of some kind, a search term is "allometry". Jan 23 '15 at 13:02
• stats.stackexchange.com/questions/59784/… is one place to start. Jan 23 '15 at 13:15