Resultados inferenciales: análisis de variables y dimensiones Prueba de Normalidad

In the example given in Table 11.1 , the fact that the vessels are also grouped into replica-tions, 1 complete replication for each of the 10 occasions, gives a more complex model.

Using the commonly used notation to describe the basic model of an ANOVA fi rst described in Statnote 10 , the two - way design includes a term for the replication effect b in addition to the treatment effect a , namely

xij= + + +μ ai bj eij (11.1) Hence, the ANOVA table (Table 11.2 ) includes an extra term for replications, that is, the variation between occasions. In the terminology used in Statnote 10 , treatment a i is a fi xed - effect factor whereas blocks or occasions b j are a random - effect factor. In addition to the assumptions made in the randomized design, namely homogeneity of variance, additive class effects, and normal distribution of errors, this type of design makes the additional assumption that the difference between treatments is consistent across all rep-lications (Snedecor & Cochran, 1980 ).

11.4.3 Interpretation

The ANOVA appropriate to the two - way design is shown in Table 11.2 . This design is often used to remove the effect of a particular source of variation from the analysis. For example, if there was signifi cant variation due to replications and if treatments had been allocated to vessels at random, then all of the between replicates variation would have been included in the pooled error variance. The effect of this would be to increase the

66 TWO-WAY ANALYSIS OF VARIANCE

error variance and to reduce the P ′ of the experiment (see Statnote 9 ), thus making it more diffi cult to demonstrate a possible treatment effect. In a two - way design, however, varia-tion between replicavaria-tions attributable to occasions is calculated as a separate effect and, therefore, does not appear in the error variance. This may increase the P ′ of the experiment and make it more probable that a treatment effect would be demonstrated. In the example quoted (Table 11.2 ), there is a highly signifi cant effect of media supplement ( F = 27.32, P < 0.001). In a two - way design, planned comparisons between the means or post hoc tests can be performed as for the randomized design (see Statnote 7 ). Hence, Scheff é ’ s post hoc test (see Statnote 7 ) suggested that this result is largely due to the effect of supple-ment S 1 increasing yield. In addition, in a two - way design, the variation due to replications is calculated ( F = 2.39), and this was not quite signifi cant at P = 0.05. The borderline signifi cance suggests there may have been some differences between replications, and removing this source of variation may have increased the P ′ of the experiment to some degree. This statnote also illustrates an alternative method of increasing the P ′ of an experi-ment, namely by altering its design.

A comparison of the ANOVA table in Table 11.1 with that for a one - way ANOVA in a randomized design demonstrates that reducing the error variance by “ blocking ” has a cost, namely a reduction in the DF of the error variance, which makes the estimate of the error variation less reliable. Hence, an experiment in a two - way design would only be effective if the blocking by occasion or some other factor reduced the pooled error vari-ance suffi ciently to counter the reduction in DF (Cochran & Cox, 1957 ; Snedecor &

Cochran, 1980 ).

11.5 CONCLUSION

The two - way design has been variously described as a matched - sample F test, a simple within - subjects ANOVA, a one - way within - groups ANOVA, a simple correlated - groups ANOVA, and a one - factor repeated measures design! This confusion of terms is likely to lead to problems in correctly identifying this analysis within commercially available packages. The essential feature of the design is that each treatment is allocated by ran-domization to one experimental unit within each group or block. The block may be a plot of land, a single occasion in which the experiment was performed, or a human subject. The blocking is designed to remove an aspect of the error variation and increase the P ′ of the experiment. If there is no signifi cant source of variation associated with the blocking, then there is a disadvantage to the two - way design because there is a reduction in the DF of the error term compared with a fully randomized design, thus reducing the P ′ of the analysis.

Statnote 12

TWO - FACTOR ANALYSIS OF VARIANCE

The factorial experimental design.

Interactions between variables.

Statistical model of a factorial design.

Partitioning the treatments sums of squares into factorial effects (contrasts).

12.1 INTRODUCTION

The analyses of variance (ANOVA) described in previous statnotes (see Statnotes 6 , 10 , and 11 ) are examples of single - factor experiments. The single - factor experiment comprises two or more treatments or groups often arranged in a randomized design, that is, experi-mental units or replicates are assigned at random and without restriction to the treatments.

An extension to this experimental design is the two - way design (see Statnote 11 ) in which the data are classifi ed according to two criteria, that is, treatment or group and the replicate or block to which the treatment belongs. A factorial experiment, however, differs from a single - factor experiment in that the effects of two or more factors or “ variables ” can be studied at the same time. Combining factors in a single experiment has several advantages.

First, a factorial experiment usually requires fewer replications than an experiment that studies each factor individually in a separate experiment. Second, variation between treat-ment combinations can be broken down into components representing specifi c compari-sons or contrasts (Ridgman, 1975 ; Armstrong & Hilton, 2004 ) that reveal the possible

Statistical Analysis in Microbiology: Statnotes, Edited by Richard A. Armstrong and Anthony C. Hilton Copyright © 2010 John Wiley & Sons, Inc.

68 TWO-FACTOR ANALYSIS OF VARIANCE

synergistic or interactive effects between the factors. The interactions between factors often provide the most interesting information from a factorial experiment and cannot be obtained from a single - factor experiment. Third, in a factorial design, an experimenter can often add variables considered to have an uncertain or peripheral importance to the design with little extra effort. This statnote describes the simplest case of a factorial experiment incorporating two factors each present at two levels.

12.2 SCENARIO

The kitchen dishcloth is increasingly recognized as a primary reservoir of bacteria with potential to cause widespread cross contamination in food preparation environments. An investigator wished to study the infl uence of the material from which the dishcloth was manufactured (cloth or sponge) (factor A ) and the effect of rinsing the dishcloth in running water (factor B ) on the number of bacteria subsequently transferred to a food preparation surface (Hilton & Austin, 2000 ). Dishcloths of each material type were inoculated with 1 ml of a 10 ⁸ CFU/ml Escherichia coli culture and after 10 minutes, the cloth was wiped over an appropriate area of sterile cutting board. Additional pieces of cloth were inoculated but rinsed in sterile running water before wiping. The cutting board was subsequently swabbed to recover E. coli that had been deposited from the wiping by the cloth.

12.3 DATA

The data comprise the number of bacterial colonies obtained on nutrient agar from the two types of dishcloth, rinsed and unrinsed, and are presented in Table 12.1 . The objec-tives of the experiment were to determine whether the ability of bacteria to be transferred from the dishcloth varied, fi rst, with the type of dishcloth (factor A ) and, second, with rinsing treatment (factor B ) and whether the two factors had an independent infl uence on the numbers of bacteria. Hence, there are four treatment combinations, that is, two types of cloth each of which was either rinsed or not rinsed. This type of design is the simplest type of factorial experiment and is also known as a 2 ² factorial, that is, two factors with two levels of each factor. In this notation, the superscript refers to the number of factors or variables included and the integer the number of levels of each factor. As the number of variables included in the experiment increases, the superscript can take any integer value. Hence, a 2 ⁴ factorial would have four separate factors each at two levels.

ANALYSIS 69

12.4 ANALYSIS