types i, ii, and iii contingency tables

hereby some examples drawn from Sokal and Rohlf (1995: 724 et seq.), and edited and expanded a bit by me to (hopefully) clarify the distinction among Models I, II, and III contingency tables:

Type I: neither set of column totals set by investigator:

100 plants are examined, and their soil type and leaf texture is recorded:

	Pubescent Leaves	Smooth Leaves	Total
Serpentine Soil	12	30	42
Non-serpentine Soil	47	11	58
Total	59	41	100

Type II: one set of column totals set by investigator:

100 moths are exposed to bird predation: 50 light morphs and 50 dark morphs (note that the proportion doesn't have to be 50:50, though); investigator records whether moths are eaten or not:

	Prey	Survivor	Total
Light Morph	39	11	50
Dark Morph	30	20	50
Total	69	31	100

Type III: both sets of column totals set by investigator:

one hundred beans are placed in a jar: 50 with thick skins and 50 with thin skins (again, doesn't have to be 50:50). seventy hungry weevil larvae -- each of which will burrow in to one unoccupied bean -- are added to the jar, and some time later the investigator records the numbers of each type of bean and whether or not it was attacked:

	Attacked	Not Attacked	Total
Thick Skin	a	b	50
Thin Skin	c	d	50
Total	70	30	100

at first glance, it seems that having both the row totals and the column totals fixed will automatically fix the cell totals; this is not actually true, as the following values of a, b, c, and d will illustrate:

a = 20, b = 30, c = 50, d = 0;
a = 25, b = 25, c = 45, d = 5;
a = 35, b = 15, c = 35, d = 15;
etc.

Sokal and Rohlf indicate that they have "not yet encounted a [non-hypothetical] example of this model."

ancova

as a (relatively) uncomplicated published example of ancova, i humbly present for my biostats students' consideration the following: http://www.tulane.edu/~guill/Reprints/Guill_and_Heins_2000.pdf

perhaps most useful to them will be the formats it uses for reporting the results of the analyses (which -- looking back over it, i'm embarrassed to say, aren't perfect: one needs 2 values for the degrees of freedom for an F ratio. my bad.)

also, it may serve as a reasonably useful model for what i'll be looking for in their independent projects -- basically something approximating the 'methods' and 'results' section of this paper in length and depth, supported perhaps by a few well-crafted figures and tables, as appropriate. anything beyond that (e.g. intro or discussion) will be lagniappe.

anova by hand

and here i was thinking i was being all progressive and modern by not making my biostats students work through all the calculations for anova by hand, but -- lo and behold! -- busy tosser has opined that the old-skool approach might actually be helpful. she's probably right;) so, let it never be said that i'm not willing to hand out additional work when it's asked for -- here you go:

a quick google search on "anova by hand" turned up the following worksheet:statisticshell.com/anovabyhand.pdf. the 'parent' site that it comes from -- statisticshell.com -- is a hoot. i've worked through the worksheet and it's actually quite good -- he walks you through one example (response to viagra, no less!) and then gives you a second problem to work on your own, followed by the answers to that one as well. i checked his results in R and get the same answers as he did, so -- if you're so inclined -- have at it!

let me know if it helps.

testing testing

this is just a test to see what kind of HTML formatting blogger will allow you to use such that we can incorporate R code and output while preserving spacing for columns and the like (so it looks like the output on your R terminal).

for instance, compare this:

> head(iris)
Sepal.Length Sepal.Width Petal.Length Petal.Width Species
1 5.1 3.5 1.4 0.2 setosa
2 4.9 3.0 1.4 0.2 setosa
3 4.7 3.2 1.3 0.2 setosa
4 4.6 3.1 1.5 0.2 setosa
5 5.0 3.6 1.4 0.2 setosa
6 5.4 3.9 1.7 0.4 setosa

with this:

> head(iris)
  Sepal.Length Sepal.Width Petal.Length Petal.Width Species
1          5.1         3.5          1.4         0.2  setosa
2          4.9         3.0          1.4         0.2  setosa
3          4.7         3.2          1.3         0.2  setosa
4          4.6         3.1          1.5         0.2  setosa
5          5.0         3.6          1.4         0.2  setosa
6          5.4         3.9          1.7         0.4  setosa
>

the trick is to enclose the cut-and-pasted output from R in between the HTML markup tags <PRE> ... <\PRE>

g&e ch 6

i realized only this morning that i hadn't blogged my usual preview note for the upcoming chapter. i apologize; theryn will be acting as MC today, leading our class discussion, and i guess i shifted into "participant" mode a little too soon. at any rate, it's probably too late for anyone to benefit from reading my rambling thoughts (i doubt anyone is up reading my blog at 6 AM) but at least this will be here for later review and reflection.

this chapter is straightforward, mostly non-quantitative (only a couple of equations pertaining to what is -- imho -- a largely tangential bit about modelling...), and full of good advice on how to worry about all of the things that might go wrong with your field study. no, not really... but it does lay out and emphasizes well the general principles of replication and randomization, and how necessary they are in order to make the results of your hard work as generalizable (and interesting, and useful, and, therefore, publishable) as possible.

i think the authors may be a bit too sanguine about the about the possibility of controlling for or taking into account all of the potential confounding variables that may affect a field study. my perspective is that, given finite resources and time, there will always be the risk of an unmeasured covariate that presents itself as possibly important after-the-fact, but that a reasonably well replicated and randomized design minimizes (but doesn't eliminate) this risk.

G&E CH 5

I like the framework that G&E have laid out in this chapter on the several different general approaches to statistical analysis, and I do think it is all worth reading fairly closely. That said, I think the simple example that they use (ant nests in forests and fields) to illustrate the different approaches (an excellent pedagogical approach, IMHO) is telling: Their descriptions of how one would go about implementing their "monte carlo" approach is clear and I expect would be easy (if tedious) for most any one at your level to implement using a spreadsheet. Their description of the standard parametric analysis is -- I think -- a reasonable compromise between overview and detail (which you'll get a a little later in the semester); after reading it I think you should have some sense of what F represents in an ANOVA (although not the ability to calculate it yet). As to Bayesian analysis -- I'll keep my opinion to myself for now, but I will prompt you with the following: after reading through this section, ask yourself if you could begin to put together the approach that you would need to follow in order to repeat the authors' analysis.

I do think they do a bit of a disservice to non-parametric statistics, and, given their ubiquity, maybe should have spent a bit more time on them. We will, ultimately, come back to some of the more popular of these approaches (e.g. chi square) in later chapters.

JV ch. 4

although this chapter is titled 'multivariate data', most of it is spent filling in the gaps and expanding your understanding of how R deals with data in the form of lists and data frames. [although we haven't really talked about it, you've been using data frames since you first started using attach().] also of note will be the additional practice you will get (and skills you will develop) in making plots. although it may seem insanely hard at first, once you get the hang of it, R will allow you to make some really nice plots with comparably little effort (at least in comparison to at least some other statistical graphing packages that i'm familiar with).

as to what to focus on -- at the beginning of the chapter the author again spends some time showing you how to make various tables, which, as i've indicated before, i think may be something better left to spreadsheets. at least at the beginning.

the end of section 4.1 gives you a nice explanation of high- versus low- level plotting features, and some examples of additional plotting options.

section 4.2 is a tedious but useful (and necessary) breakdown of some of the details of data frames and lists, whereas section 4.3 is, in my opinion, a little on the tangiential side. if you're reading along about xtabs(), split(), and stack(), and you're zoning out, don't worry too much. you can come back to these things when you find a problem that necessitate them.

lattice graphics (section 4.4) are pretty cool when your data are appropriate to be shown in this fashion, so this section is worth a read, whereas, possibly with the exception of 'factors', most of section 4.5 can be safely skimmed or skipped at this point (as JV himself indicates).