Treatment, ANOVA

Data treatment, ANOVA and other statistical approaches are not discussed, although of importance in reproducibility studies, where different conditions prevail like different instruments or several technicians. 

An analysis of variance can be extended to systems involving more than a single variable. For example, a two-way ANOVA can be used in a collaborative study to determine the importance to an analytical method of both the analyst and the instrumentation used. The treatment of multivariable ANOVA is beyond the scope of this text, but is covered in several of the texts listed as suggested readings at the end of the chapter. 

The dry weights (104 C, 48 hr) of ten plants from each treatment group were taken at the termination of each experiment in order to compare growth effects with plant water status. Dry weight data were analyzed using analysis of variance (ANOVA) and Duncan s multiple-range test. Diffusive resistance and water potential were evaluated using the t-test. Each of these and subsequent experiments was replicated. 

The wheat bran used in these studies was milled for us from a single lot of Waldron hard red spring wheat. Other foods and diet ingredients were purchased from local food suppliers. Data from HS-I was analyzed statistically by Student s paired t test, each subject acting as his own control. A three-way analysis of variance (ANOVA) was performed to test for significant differences betwen diet treatments, periods and individuals in HS-II and HS-III. 

There are various other ways of examining the variate in question in this case. Let us first examine simple one-way ANOVA of the variate by sex as in Table 16.16. In neither of the two cases was there any indication of significant treatment differences at any reasonable level. Because the two sexes did not show any pretreatment differences based on the two-factor analysis of the covariate, let us combine the two sexes and analyze the data by one-way ANOVA as in Table 16.17. In this case, because of the increased sample sizes for combining the two sexes, there was indication of some treatment differences (p = 0.0454). Unfortunately, this analysis assumes that because there were no pretreatment differences between the two sexes, that pattern will hold during the posttreatment period. That often may not be the case because of biological reasons. 

In other words, we have gained about 2.3-fold precision by ANCOVA over ANOVA in resolving treatment effect. 

Data obtained from animal experiments were expressed as mean standard error ( SEM). Statistical differences between the treatments and the control were evaluated by ANOVA and Students-Newman-Keuls post-hoc tests, p < 0.05 was considered to be significant ( p < 0.05 p < 0.01 p < 0.001). 

Statistical Analysis. Analysis of variance (ANOVA) of toxicity data was conducted using SAS/STAT software (version 8.2 SAS Institute, Cary, NC). All toxicity data were transformed (square root, log, or rank) before ANOVA. Comparisons among multiple treatment means were made by Fisher s LSD procedure, and differences between individual treatments and controls were determined by one-tailed Dunnett s or Wilcoxon tests. Statements of statistical significance refer to a probability of type 1 error of 5% or less (p s 0.05). Median lethal concentrations (LCjq) were determined by the Trimmed Spearman-Karber method using TOXSTAT software (version 3.5 Lincoln Software Associates, Bisbee, AZ). 

All the data from the experiments were analyzed through an ANOVA program for a randomized complete bloc)c design (17) and comparisons were made between treatments. The percentages of inhibition were calculated by considering the control as zero. 

In this setting there is a technique, termed one-way analysis of variance (one-way ANOVA), which gives an overall p-value for the simultaneous comparison of all of the treatments. Suppose, for example, we have four treatment groups with means Pi, p2> P 3 P4. This procedure gives a p-value for the null hypothesis ... 

Statistical analysis proceeds through two-way analysis of variance (ANOVA). The focus in this methodology is to compare the treatment groups while recognising potential centre differences. To enable this to happen we allow the treatment means and gig to be different in the different centres as seen in Table 5.2. 

This null hypothesis is saying that the treatment means are the same within each centre, but not necessarily across centres we are allowing the centres to be different. Two-way ANOVA gives us a p-value relating to this null hypothesis. If that p-value is significant (p < 0.05) then we reject the null hypothesis and there is evidence that the treatments are different. 

Initially you may think that this approach to comparing the treatments is over elaborate why not just compare the overall means (which happened to be 11.8 and 4.9) in an unpaired t-test Well in one sense you could, but it would not be the most efficient thing to do. Simply comparing the overall means loses the fact that the means in centre 1,12.4 and 5.8, are linked, as are those in each of the other centres. The two-way ANOVA procedure maintains that link, it simultaneously compares... 

We saw in the previous chapter how to account for centre in treatment comparisons using two-way ANOVA for continuous data and the CMH test for binary, categorical and ordinal data. These are examples of so-called adjusted analyses, we have adjusted for centre differences in the analysis. 

Suppose that we have just two centres (and two treatment groups). If we define binary indicators, say z and x, to denote treatment group and centre, respectively, then including z and x in an ANCOVA is identical mathematically to the corresponding two-way ANOVA. This connection is true more generally. If we were now to add more centres (say four in total) to the ANOVA then defining binary indicators to uniquely define these centres Xj = 1 for a patient in centre 1, Xj = 1 for a patient in centre 2, X3 = 1 for a patient in centre 3 with 0 values otherwise, then ANCOVA with terms z, Xj, Xj and X3 would be mathematically the same as ANOVA. We would obtain the same p-values, (adjusted) estimates of treatment effect, confidence intervals etc. 

This link applies also to the p-value from the unpaired t-test and the confidence interval for p, the mean difference between the treatments, and in addition extends to adjusted analyses including ANOVA and ANCOVA and similarly for regression. For example, if the test for the slope b of the regression line gives a significant p-value (at the 5 per cent level) then the 95 per cent confidence interval for the slope will not contain zero and vice versa. 

The appropriate test when comparing more than two means is analysis of variance (ANOVA). The essential process in ANOVA is to split up, or decompose, the overall variance in the data. This variability is due to differences between the means due to the treatment effect (between-group variance) and that due to random variability between individuals within each group (within-group variance, sometimes called unexplained or residual variance), hence the name analysis of variance. ... 

The constriction amplitude of the light reflex differed significantly among the treatment conditions (Figure 7.2). A two-way ANOVA indicated significant differences among drug conditions... 

Table 2 Increase in growth and decrease in root-knot nematode (Meloidogyne incognita) infestation of cowpea plants following treatment with foliar spray of the aqueous solution of Cina 1000c. Values are means with S.E. of 10 plants of each batch. Different letters (a, b, c) against the values in a column indicate significant difference (P<0.01) by ANOVA. Data from Environ Ecol 17 269-273 (1999) with permission.

Figure 8. Stomatal density (SD A), epidermal cell density (ED B) and stomatal index (SI C) of Quercus kelloggii leaves grown in growth chambers under low temperature (day 20°C, night 15°C) and high temperature regimes (day 27°C, night 22°C). Measurements for low temperature are based on eleven leaves (seven random counts each) from six trees and for the high temperature on nine leaves from five trees. Nested mixed-model ANOVA indicates no significant differences between low and high temperature treatments for SD (0.323), ED (0.442) or SI (0.313). Error bars represent 1 S.E.M.

Results were analyzed by nested mixed-model ANOVA s using general linear procedures, in the MINITAB 15 statistical program. Nested mixed-model ANOVA was used when multiple leaves per tree and multiple trees per treatment were available. Additional analyses were linear and quadratic regressions (performed in MINITAB 15 and Excel), and when significant differences occurred, means were compared using Student s t-test or nested mixed-model ANOVA. 

Fig. 14.3. Treatment effect on tumor growth. (A) Treatment with IFL significantly (P= 0.0009, RM ANOVA) inhibited the growth of FIT-29LP tumors as measured by bioluminescence imaging. (B) Tumor weight measurements confirmed antitumor effects of IFL. Mean tumor weight in the IFL group was significantly (P< 0.0001, one-way ANOVA) lower...

Perform statistical analysis utilizing repeated measures ANOVA (JMP software, Version 5 from SAS Institute, Cary, NC) to evaluate the treatment effects on tumor growth. Compare tumor weight measurements between treatment groups utilizing one-way ANOVA. 

FIGURE 10.2 (CONTINUED) and unattached larvae. N = six (6) replicates (dishes) were done for all treatments. The results of the assay are expressed as percentage settlement of the seawater (untreated) control. Data are mean + S.E. Treatments lacking error bars indicate 100% settlement in all replicates. Statistical analysis of the data (separate one-factor analysis of variance ANOVA for each of Figure 10.2A and 10.2B, followed by Tukey s post-hoc comparison among means) showed that only extracts from D. pulchra significantly deterred settlement (at both natural and twice natural concentrations). 

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