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.
When she collected the data, she did not have equal numbers of data
(measured diameters) for all levels. I provide the count of each dataset
in the table below (the table has more meaning if your nntp client uses a
fixed pitch font):
Counts of data (n) for each dataset:
Initial Plating Density
--------------------------------------------
Days of | 10/cm2 | 100/cm2 | 1000/cm2 |
Incubation | | | |
--------------+---------------+--------------+-------------+
| | | |
2 | n = 0 | n = 11 | n = 0 |
| | | |
--------------+---------------+--------------+-------------+
| | | |
5 | n = 22 | n = 38 | n = 31 |
| | | |
--------------+---------------+--------------+-------------+
| | | |
9 | n = 19 | n = 142 | n = 72 |
| | | |
--------------+---------------+--------------+-------------+
As you can see, in some cases she did not get any data at all for a
particular treatment (2-days at 10/ and 1000/cm2), and in other cases, she
outdid herself in collecting more diameters than she really need (day 9 at
100/cm2, and probably 1000/cm2).
I cranked through the 2-way ANOVA on my Excel spreadsheet (I did not use
Microsoft's bundled analytical tool, and I don't have SPSS or other
handholding tool) and got these:
===============================================================
Source of
Variation SS df MS F p
------------ ---------- ----- -------- --------- ---------
Densities 46.937 2 23.48 0.255 0.775
Days 11.828 2 5.914 0.0642 0.938
Days x Densities 16.688 2 8.344 0.0906 0.913
Error 30196.541 328 92.06
Total 39189.826 334
===============================================================
This is without having tested the assumptions required for ANOVA in
general. With the grossly unequal sample sizes, it is clearly essential to
test for normality and variance homogeneity. These are my tests for
variance homogeneity:
===============================================================
Levene's
SS df MS F p
------- ----- -------- -------- ----------
between 1170.5 6 195.1 5.59 0.000015
within 11450.0 328 34.91
total 12620.6 334
===============================================================
===========================================================================
Bartlett's
2d-10 5d-10 5d-100 5d-1000 9d-10 9d-100 9d-1000
------ ------- ------- -------- ------- -------- ---------
SS[i] 344.10 2421.69 3063.46 714.20 2009.04 18539.90 3104.15
df 10 21 37 30 18 141 71
df * ln(VAR) 35.38 99.70 163.41 95.10 84.87 687.93 268.22
1 / df 0.10 0.05 0.03 0.03 0.06 0.01 0.01
pooled Var 92.06
LN(pooled Var) 4.52
B 48.76 (Bartlett's T statistic numerator)
C 1.03 (Bartlett's T statistic denominator)
T 47.28 (Bartlett's T statistic)
chi-square [T,3] p <<<< 0.0001
===========================================================================
These after throwing out one outlier.
I don't have Sokal & Rohlf's stat tables appendix to look up NED values for
Q-Q plots to check for normality, so got stuck there.
My grad student input her data into SPSS (for Win) and its analysis
reported that there was a significant difference in the factor for initial
plating density (something she was hoping for), and no sig diff for
anything else, but SPSS did not report what "correction" it used.
My question:
What 2-Way ANOVA tool do I use for exploring differences in these
treatments if the variances between groups is clearly not equal
(homogeneous)?
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.
Thanks.
.
.
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