Experimental design replication

Scanning electron microscopy and replication techniques provide information concerning the outer surfaces of the sample only. Accurate electron microprobe analyses require smooth surfaces. To use these techniques profitably, it is therefore necessary to incorporate these requirements into the experimental design, since the interfaces of interest are often below the external particle boundary. To investigate the zones of interest, two general approaches to sample preparation have been used. 

Soybean bloassays of root exudates. Four soybean seeds ( Bragg ) were planted In each of 100 12.5 cm plastic pots filled with an artificial soil mix consisting of perlite/coarse sand/coarse vermiculite 3/2/1 by volume. After one week the plants were thinned to two per pot and the treatments were begun. The experimental design was a completely randomized design with 10 replications (pots) per treatment. On the first day of each week each pot was watered with 300 ml effluent from the appropriate growth units. On the fifth day of each week all pots were watered with Peter s Hydro-sol solution with CaCNOj. At other times the pots were watered as needed with tap water. On the second and fifth day of each week the height of the soybeans (base to apical bud) was measured. 

Root elongation bloassay of root exudates. Five ml aliquots of the root exudates were pipetted onto three layers of Anchor1 germination paper In a 10 by 10 by 1.5 cm plastic petri dish. Twenty five radish or tomato seeds were placed in a 5x5 array in each petri dish. Radish seeds were incubated at 20C for 96 hours tomato seeds were incubated at 20C for 168 hours, before the root length was measured. Experimental design was a completely randomized design with three replications (dishes) per treatment per bioassay seed species. The bioassay was repeated each week for 23 weeks. 

Statistical experimental design is characterized by the three basic principles Replication, Randomization and Blocking (block division, planned grouping). Latin square design is especially useful to separate nonrandom variations from random effects which interfere with the former. An example may be the identification of (slightly) different samples, e.g. sorts of wine, by various testers and at several days. To separate the day-to-day and/or tester-to-tester (laboratory-to-laboratory) variations from that of the wine sorts, an m x m Latin square design may be used. In case of m = 3 all three wine samples (a, b, c) are tested be three testers at three days, e.g. in the way represented in Table 5.8 ... 

This efficient statistical test requires the minimum data collection and analysis for the comparison of two methods. The experimental design for data collection has been shown graphically in Chapter 35 (Figure 35-2), with the numerical data for this test given in Table 38-1. Two methods are used to analyze two different samples, with approximately five replicate measurements per sample as shown graphically in the previously mentioned figure. 

Current practice in microarray experimentation suggests that a balance design with adequate replication be used. Good experimental design and execution will produce data that minimize technical variance, allowing the statistical analyses to evaluate biological variance more effectively Still, the nature of the data requires that an estimate of the FDR be included in the statistical analysis. This enables the researcher to assess the reliability/validity of the results of the statistical analysis. As discussed earlier, cDNA microarray... 

The estimation of purely experimental uncertainty is essential for testing the adequacy of a model. The material in Chapter 3 and especially in Figure 3.1 suggests one of the important principles of experimental design the purely experimental uncertainty can be obtained only by setting all of the controlled factors at fixed levels and replicating the experiment. 

We will also assume that we have a prior estimate of for the system under investigation and that the variance is homoscedastic (see Section 3.3). Our reason for assuming the availability of an estimate of is to obviate the need for replication in the experimental design so that the effect of the location of the experiments in factor space can be discussed by itself. 

If we assume the model y, = 0 + r, , the data and uncertainties are as shown in Figure 8.1. We can test this model for lack of fit because there is replication in the experimental design which allows an estimate of the purely experimental uncertainty (with two degrees of freedom). 

Given these five levels of pH, how can the five replicate experiments be allocated One way is to place all of the replicates at the center factor level. Doing so would give a good estimate of at the center of the experimental design, but it would give... 

Let us try instead an experimental design in which two replicates are carried out at the center point and three replicates are carried out at each of the extreme points (-2 and +2). Then... 

Inspection of the coded experimental design matrix shows that the first four experiments belong to the two-level two-factor factorial part of the design, the next four experiments are the extreme points of the star design, and the last four experiments are replicates of the center point. The corresponding matrix for the six-parameter model of Equation 12.54 is... 

The lower left panel in Figure 13.2 shows the central composite design in the two factors X, and X2. The factor domain extends from -5 to +5 in each factor dimension. The coordinate axes in this panel are rotated 45° to correspond to the orientation of the axes in the panel above. Each black dot represents a distinctly different factor combination, or design point. The pattern of dots shows a central composite design centered at (Xj = 0, Xj = 0). The factorial points are located 2 units from the center. The star points are located 4 units from the center. The three concentric circles indicate that the center point has been replicated a total of four times. The experimental design matrix is... 

Replicates don t all have to be carried out at the center point. Using the experimental design of Figure 13.2 as a basis. Figures 13.9 and 13.10 show the effects of different distributions of replicates. 

In Figure 13.9, instead of carrying out four replicate experiments at the center point (as in Figure 13.2), the four replicates are carried out such that one experiment is moved to each of the existing four factorial points. The experimental design matrix is... 

Figure 13.10 shows the effect of placing the four replicates at each of the star points. The experimental design matrix is... 

In this chapter we explore factorial-based experimental designs in more detail. We will show how these designs can be used in their full factorial form how factorial designs can be taken apart into blocks to minimize the effect of (or, if desired, to estimate the effect of) an additional factor and how only a portion of the full factorial design (a fractional replicate) can be used to screen many potentially useful factors in a very small number of experiments. Finally, we will illustrate the use of a Latin square design, a special type of fractionalized design. 

The complete design is seen in the score space with replicate center points clearly visible. Note that the interpretation of scores plots is not always as straightforward as in this example. The experimental design is not seen if the experiment is not well designed or if the problem is high dimensional. The level of impEcidy modeled components (e.g., component O also has an effect on the relative position of the samples in score space. For this example, the effect of C on the relative placement of the samples in score space is small. 

Scores Plot (Sample Diagnostic) Figure 5-132 displays the Factor 2 versus Factor l scores for the MCB model (showing 98.41% of the spectral variance). The experimental design for this data set is not readily discernible because this plot shows only two dimensions. However, samples 1-4 define the extremes, which makes sense because these are the pure spectra. As expected, the center-point replicates lie very near each other and are in the middle of the plot. 

Figure 1 provides adjustments to critical values for CV p when a method is biased. The dotted curve gives critical values of CV-p as a function of bias for a statistical significance test performed at the 5% probability level. Because uniform replicate determinations of the bias were not made in the validation tests, the bias is treated as a known constant rather than an estimated value. The experimental design could be modified to permit determination of the imprecision in the bias by providing for uniform replication of the independent method as well as the method under evaluation. Then the decision chart could be modified to include allowance for variability of replicate bias determinations. 

We have presented a statistical experimental design and a protocol to use in evaluating laboratory data to determine whether the sampling and analytical method tested meets a defined accuracy criterion. The accuracy is defined relative to a single measurement from the test method rather than for a mean of several replicate test results. Accuracy here is the difference between the test result and the "true value, and thus, must combine the two sources of measurement error ... 

Fractional replicates of experimental designs in which all factors are at the same number of levels can be partially replicated in fractions whose denominators are multiples of the number of levels. These designs are the so-called Latin square designs. 

The data for this example are taken from ref. 26 in the bibliography. Many experimental designs are available but a simple full factorial is taken by way of example. A full factorial design is where all combinations of the factors are experimentally explored. This is usually limited from practical consideration to low values. To simplify the matter further no replication was used. 

Any experimental design that is intended to determine the effect of a parameter on a response must be able to differentiate a real effect from normal experimental error. One usual means of doing this determination is to run replicate experiments. The variations observed between the replicates can then be used to estimate the standard deviation of a single observation and hence the standard deviation of the effects. However, in the absence of replicates, other methods are available for ascertaining, at least in a qualitative way, whether an observed effect may be statistically significant. One very useful technique used with the data presented here involves the analysis of the factorial by using half-normal probability paper (19). 

---