VIII. PROPUESTA 1. Datos generales:
10. Evaluación, monitoreo y control
Using the notation described previously (see Statnote 10 ), this design can be described as follows:
xijk = + + +μ ai bj ( )ab ij+eijk (12.1) In this model, x ijk is the value of the k th replicate of the i th level of factor A and the j th level of factor B , a i and b j are the main effects of each of the two factors, and ( ab ) ij represents the two - factor interaction between A and B .
As in previous examples, the total sums of squares (SS) of the data can be partitioned into components associated with differences between the effects of the cloth and rinsing treatment (see Statnote 7 ). In this case, the between - treatments SS can be partitioned into contrasts that describe the “ main ” effects of A and B and the “ interaction ” effect A × B . These effects are linear combinations of the treatment means, each being multiplied by a number or coeffi cient to calculate a particular effect. In fact, the meaning of a factorial effect can often be appreciated by studying these coeffi cients (Table 12.2 ). The effect of dishcloth is calculated from those replicates representing the cloth data ( + ) compared with those that represent the sponge data ( − ). Note that in a factorial design, every observation is used in the estimate of the effect of every factor. Hence, factorial designs have internal replication and this may be an important consideration in deciding the number of replica-tions to use. Each replicate is actually providing two estimates of the difference between cloths and sponges. The main effect of rinsing is calculated similarly to that of dishcloth.
By contrast, the two - factor interaction (dishcloth × rinsing) can be interpreted as a test of whether cloth type and rinsing treatment act independently of each other. A signifi cant interaction term would imply that the effect of the combination of cloth type and rinsing treatment would not be predictable from knowing their individual effects; a common occurrence in many real situations in which it is a combination of factors that determine a particular outcome.
In a 2 2 factorial, partitioning the treatments SS into factorial effects provides all the information necessary for interpreting the results of the experiment and further post hoc tests would not be necessary. With more complex factorial designs, for example, those with more than two levels of each factor, further tests may be required to interpret a main effect or an interaction. With factorial designs, however, it is better to defi ne specifi c comparisons before the experiment is carried out rather than to rely on post hoc tests that
TA B L E 12.2 Coeffi cients for Calculating Factorial Effects or Contrast in a 2 2 Factorial for the
70 TWO-FACTOR ANALYSIS OF VARIANCE
compare all possible combinations of the means. Factorial experiments can be carried out in a completely randomized design (see Statnote 6 ), in randomized blocks (see Statnote 11 ), or in more complex designs (Cochran & Cox, 1957 ). The relative advantages of these designs are the same as for the one - way design.
12.4.2 Interpretation
The resulting ANOVA (Table 12.3 ) is more complex than in a one - way experiment because the between - groups or treatments SS is partitioned into three factorial effects, namely the main effects of dishcloth type and rinsing and the interaction between the two factors. In the present example, there is a main effect of dishcloth type ( F = 20.99, P < 0.01) and of rinsing ( F = 28.92, P < 0.001), suggesting signifi cantly more bacteria were transferred from the cloth than the sponge and signifi cantly fewer after rinsing both materials. In addition, there is signifi cant interaction between the factors ( F = 20.97, P < 0.01), suggesting that the effect of rinsing is not consistent for the two types of dish-cloth, namely rinsing has less effect on the number of bacteria transmitted from the sponge compared with the cloth. Examination of the treatment means suggests that the sponge transfers a smaller proportion of its bacterial load to the food preparation surface compared with the cloth. This effect is probably attributable to organisms being more exposed on the surface of the cloth and therefore more liable to be transferred compared with the more cavernous sponge (Hilton & Austin, 2000 ).
12.5 CONCLUSION
Experiments combining different groups or factors are a powerful method of investigation in microbiology. ANOVA enables not only the effects of individual factors to be estimated but also their interactions, information that cannot be obtained readily when factors are investigated separately. In addition, combining different treatments or factors in a single experiment is more effi cient and often reduces the number of replications required to estimate treatment effects adequately. Because of the treatment combinations used in a factorial experiment, the DF of the error term in the ANOVA is a more important indica-tor of the P ′ of the experiment than simply the number of replicates per treatment (see Statnote 9 ).
TA B L E 12.3 Analysis of Variance of Dishcloth Data in Table 12.1 a
Source DF SS MS F P
Cloth/sponge 1 2002.83 2002.83 20.99 < 0.01
Rinsing 1 2760.42 2760.42 28.92 < 0.001
Interaction 1 2001.33 2001.33 20.97 < 0.01
Error 8 763.49 95.44
a DF = degrees of freedom, SS = sums of squares, MS = mean square, F = variance ratio, P = Probability.
Statnote 13
SPLIT - PLOT ANALYSIS OF VARIANCE
Major and minor factors.
Statistical model of the split - plot design.
How to identify a split - plot design.
Disadvantages of a split - plot design.
13.1 INTRODUCTION
In Statnote 12 , an investigator wished to study the infl uence of type of dishcloth (cloth or sponge) (factor A) and rinsing treatment (factor B) on the number of bacteria transferred to a food preparation surface (Hilton & Austin, 2000 ). The objectives of this experiment were to determine whether the risk of transferring bacteria from the dishcloth varied with dishcloth type, rinsing treatment, and whether the two factors had an independent infl uence on the number of bacteria transferred. Hence, there were four treatment combinations, namely, two types of cloth, each of which was either rinsed or not rinsed. This type of design is an example of the simplest factorial experiment, also known as a 2 2 factorial, that is, two factors were present with two levels of each factor. An important feature of this experimental design is that the replicates were assigned at random to all possible combinations of the two factors “ without restriction. ” In some experimental situations, however, the two factors are not equivalent to each other, and replicates cannot be assigned at random to all treatment combinations. A common case, called a split - plot design , arises when one factor can be considered to be a major factor and the other a minor factor.
Statistical Analysis in Microbiology: Statnotes, Edited by Richard A. Armstrong and Anthony C. Hilton Copyright © 2010 John Wiley & Sons, Inc.
72 SPLIT-PLOT ANALYSIS OF VARIANCE
13.2 SCENARIO
A microbiologist was interested in the number of zoospores produced by the pathogenic aquatic fungus Saprolegnia diclina (Smith et al., 1984 ). The number of zoospores produced and their motility are markedly affected by such factors of the aquatic envi-ronment as pH, oxygen tension, and the presence of biocides. To examine the effect of pH on zoospore production and motility, parent colonies of S. diclina were placed at the end of experimental counting channels each consisting of fi ve sequential chambers (A to E) (Fig. 13.1 ). The channels were fi lled with sterile 5 μ m phosphate buffer and then modifi ed to provide environments of either pH = 5.0 or 7.0. Zoospore activity was determined by counting the number of encysted zoospores within each of the fi ve chambers, representing different distances from the parent colony. Each pH channel was replicated three times.
13.3 DATA
The data are presented in Table 13.1 . This experiment also has two factors similar to Statnote 12 , namely, variation in pH (5.0 or 7.0) and the distance the zoospores travel along the channel from the parent colony (fi ve positions being sampled from A to E). The problem that arises in this type of design is the dependence or correla-tion between the measurements made in the different chambers within the same channel. Hence, in this experiment, pH is regarded as the major factor being applied to the channel as a whole, whereas distance along the channel is the minor factor, representing the chambers or subdivisions of the channel. The obvious difference between this and an ordinary factorial design is that, previously, all treatment com-binations were assigned at random to replicates, whereas in a split - plot design rep-licates can only be assigned at random to the main - plot factor, namely, the channels and not to channel – chamber combinations. In some split - plot designs, experimenters allocate replicates to major factors at random and then assign the levels of the minor factor at random within each major block. In yet other variations of this design, the subplots may be divided further to give a split – split - plot design (Snedecor & Cochran, 1980 ).
Figure 13.1. Counting chamber used to study the infl uence of pH on the production and motil-ity of zoospores (number of encysted zoospores mm 2 /24 hours) of the aquatic fungus Saprolegnia diclina .
Chambers
A B C D E
Phosphate buffer Position of colony
ANALYSIS 73
TA B L E 13.1 Infl uence of pH on Production and Motility of Zoospores (Number of Encysted Zoospores mm 2 /24 hours) of Aquatic Fungus Saprolegnia diclina
Major Factor (Channels) Minor Factor plots (channels) will be different from that between subplots (chambers). For example, one might expect there to be less natural variation between chambers within a channel than between different channels.
The resulting ANOVA (Table 13.2 ) is more complex than that for a simple factorial design (Statnote 12 ) because of the different error terms. Hence, in a two - factor, split - plot ANOVA, two errors are calculated, the main - plot error is used to test the main effect of pH while the subplot error is used to test the main effect of chambers and the possible interaction between the two factors.
74 SPLIT-PLOT ANALYSIS OF VARIANCE
13.4.2 Interpretation
There is a signifi cant increase in zoospore production at pH 7.0 compared with pH 5.0 and a marked decline in numbers of zoospores with distance from the parent colony. In addition, there is a signifi cant interaction term, suggesting that the decline in zoospores with distance down the channel is more marked at pH = 7.0 compared with pH = 5.0 (Fig.
13.2 ). The subplot error is usually smaller that the main - plot error and also has more DF.
Hence, such an experimental design will usually estimate the main effect of the subplot factor and its interaction with the main factors more accurately than the main effect of the major factor. Some experimenters will deliberately design an experiment as a split - plot to take advantage of this property.
A disadvantage of a split - plot design is that, occasionally, the main effect of the major factor may be large but not signifi cant, while the main effect of the minor factor and its interaction may be signifi cant but too small to be biologically important. In addition, a common mistake is for researchers to analyze a split - plot design as if it were a fully ran-domized two - factor experiment. In this case, the single error variance would either be too small or too large for testing the individual treatment effects, and the wrong conclusions could be drawn from the experiment. To decide whether a particular experiment is in a split - plot design, it is useful to consider the following questions: (1) Are the factors equivalent or does one appear to be subordinate to the other and especially does one rep-resent “ subdivisions ” of the other as in the chambers within a channel? (2) Is there any restriction in how replicates are assigned to the treatment combinations? (3) Is the error variation likely to be the same for each factor?
Caution should also be employed in the use of post hoc tests in the case of a split - plot design. Post hoc tests assume that the observations taken on a given channel are uncor-related so that the subplot factor group means are not uncor-related. This is unlikely since some correlation between measurements made in the different chambers of a channel is inevi-table. Standard errors appropriate to the split - plot design can be calculated (Cochran &
Cox, 1957 ; Freese, 1984 ) but should be used with caution to make specifi c comparisons
Figure 13.2. Interaction between pH and distance traveled by zoospores down a channel
( F = 75.65; P < 0.001). Error bars are 95% confi dence intervals.
CONCLUSION 75
between the treatment means. A better method is to partition the SS associated with main effects and interaction into specifi c contrasts and to test each against the appropriate error (Snedecor & Cochran, 1980 ).
13.5 CONCLUSION
In some experimental situations, the factors may not be equivalent to each other and rep-licates cannot be assigned at random to all treatment combinations. A common case, called a split - plot design, arises when one factor can be considered to be a major factor and the other a minor factor. Investigators need to be able to distinguish a split - plot design from a fully randomized design as it is a common mistake for researchers to analyze a split - plot design as if it were a fully randomized factorial experiment.