(Reply to the edstat list only. SMZ did not supply a valid e-mail address.)
Actually BOTH your postings, original and simplified, were in my INBOX this morning. As Thom remarked, ANOVA results are very dodgy in these circumstances. The first thing your client should do is to SEE what the results are trying to tell her, even though she was unable to control her sample sizes adequately. I'd illustrate, but you didn't supply cell means, let alone variances (to address one of your later questions). 1. Plot <mean diameter> (measured diameter of cultured mammalian cells) against <days of incubation> for each value of <initial plating density> This would give you three curves, one for each density, except that two of them have only two points. 2. Plot <mean diameter> against <initial plating density> for each value of <days of incubation>. This will again produce three curves, one for each incubation period, ecept that one of them will have only one point. 3. Repeat (2), using the common logarithm of <initial plating density>. (Presumably she was using powers of ten for a reason). Inspect the curves available for visible differences in slope, which would imply interaction between <plating density> and <incubation period>. For statistical evidence, (a) conduct the 2x3 complete ANOVA that is available, ignoring the lone cell at 2 days; (b) conduct regression analyses separately for each incubation period, predicting <diameter> from <density>, and choosing either raw density or its logarithm, depending on what the curves looked like; (c) especially if the curves are not parallel, conduct a regression analysis for the data of 5 and 9 days' incubation, using as predictors (1) <plating density> (or its logarithm), (2) a code for incubation period (such as "-1" for 5 days, "+1" for 9 days), (3) the product of (1) and (2), to be entered last in the regression model: the t-test for this last regression coefficient will indicate whether or not "interaction" is significant. [Aside: because the response function for <plating density> may be curved, the models in (c) should include both linear and quadratic terms in <plating density> -- there are several ways of modelling these -- and the interaction model (3) of (c) should contain two product variables. This will exhaust the possibilities. (Much more elegant if the response function is linear, not curved, in whatever function of <plating density> you end up with.)] On Fri, 12 Mar 2004, SMZ wrote in part: > I am helping a graduate student with her analysis of the diameters of > cultured mammalian cells and she is looking at the difference between > two factors: 'days of incubation' and 'initial plating density.' She > does the treatments and then measures the diameters of the cells from > images obtained from a microscope. > > She chose as 'Days of Incubation' 2, 5, and 9 days. For 'Initial > Plating Density' she chose 10 per cm2 (cm2=centimeter squared), > 100/cm2, and 1000/cm2. [snip, most of original long post] > What 2-Way ANOVA tool do I use for exploring differences in these > treatments if the variances between groups is clearly not equal > (homogeneous)? First plot the data. Then worry about possible effects of various kinds of heterogeneity. > I am reading throught Sokal & Rohlf's 3rd Edition and have trekked off > on possibly doing ANCOVA on the data, and I have yet to see what J. H. > Zar's 4th Edition will have me doing. ANCOVA? Using what for the ANOVA factor(s) and what for the covariate(s)? ------------------------------------------------------------ Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
