4. STATISTICAL METHODS
4.6 Parametric Test for Significant Difference in
Reproduction: Dunnett's Test Procedure
Dunnett's Test Procedure is used to compare the mean number
of young produced in the test sample with the mean number of
young produced in the control sample to determine if the test
sample significantly differs from the control. It is based on
the assumptions that the observations are independent and
normally distributed, and that the variance of the observations
are homogeneous across all concentrations (i.e., both the control
and test samples). Dunnett's Procedure uses a pooled estimate of
the variance which is equal to the error (within mean sum of
squares) value calculated in an analysis of variance (ANOVA).
One way to obtain an estimate of the pooled variance is to
construct an ANOVA table including all sums of squares and mean
sums of squares (Horning and Weber, 1985).
ANOVA is a technique for assessing how several nominal
independent variables affect a continuous dependent variable.
ANOVA is concerned with comparisons involving several population
means (Kleinbaum, Kupper, and Muller, 1988). However,
comparisons are made using estimates of variance. Hence, the
name ANOVA.
This study involves determining the effects of treated
wastewater effluent on the reproduction of Ceriodaohnia. Here,
the dependent (continuous) variable is the number of neonates
produced, and the independent (nominal) variable is the percent
wastewater effluent concentration.
The basic nominal variable is called a factor with different
categories of the factor referred to as levels. There are two
types of factors, fixed and random. A fixed factor is one whose
levels are the only levels of interest, as opposed to a random
factor whose levels may be regarded as a sample from some large
population of levels (Kleinbaum, Kupper, and Muller, 1988). In
this study, percent wastewater effluent concentration is a fixed
factor with two levels:
1. Control sample concentration (0% effluent)
2. Test sample concentration (percent effluent based
on instream waste concentration)
One-way ANOVA deals with the effect of a single factor
(percent wastewater effluent concentration) on a single response
variable (number of neonates produced). When the factor is
fixed, as in this case. One-way ANOVA involves a comparison of
The ANOVA table is constructed by computing the sum of
squares between sample populations (SSB), sum of squares within
sample populations (SSW), and total sum of squares (TSS). The
variability between samples (i.e., control and test sample) is
measured by SSB. It involves components of the general form
"Yi - Y" which is the difference between the "ith" sample
population mean and the overall mean. The variability within
sample populations is measured by SSW and gives no information
regarding variability between sample populations. It includes
components of the general form "Yij - Yi", which is the
difference between the "jth" observation in the "ith" sample
population. The sum of the SSB and SSW is TSS. If SSB is quite
large as compared to SSW, the majority of the total variability
is due to the differences between sample populations rather than
within sample populations.
The following formulae (Kleinbaum, Kupper, and Muller, 1988;
Horning and Weber, 1985) are used to construct an ANOVA table
and, hence, obtain an estimate of the pooled variance:
2 2
Total Sum of Squares (TSS) = summation(Yij ) - G /N
2 2Between Sum of Squares (SSB) = summation(Ti /ni) - G /N
Within Sum of Squares (SSW) = TSS - SSB
where G = Grand total number of observations (i.e.,
neonates) for all sample populations
N = Grand total number of replicates
ni = Number of replicates in sample population "i"
Ti = Total number of neonates produced in
sample population "i"
Yij = The "jth" observation for sample population "i
The ANOVA table is constructed as follows;
Source df Sum of Squares Mean Square
Between Within K-1 N-K SSB SSW MST = SSB/K-1 MSE = SSW/N-K Total N-1 TSS
where df = degrees of freedom
K = Number of sample populations MST = Between mean square
MSE = Estimate of the pooled variance
Dunnett's test statistic can now be calculated from the
following equation (Horning and Weber, 1985):
0.5
t = DMR/[MSE*(1/n1+1/n2)]
where DMR = Difference in mean reproduction between the
sample populationsUsing the data from Table 4.2, OWASA #1 Statistical
Spreadsheet Example Calculations, an example of calculating
Dunnett's test statistic is performed:
2
TSS = 4573 - (317) /24 = 385.96
2 -
SSB = 30976/12 + 19881/12 - (317) /24 = 51.04
SSW = 385.96 - 51.04 = 334.92
Completed ANOVA table for Dunnett's Procedure is shown below:
Source df Sum of Squares Mean Squares
Between 2-1=1 51.04 51.04/1=51.04
Within 24-2=22 334.92 334.92/22=15.22
Total 24-1=23 385.96
•
The mean number of neonates produced in the control sample
population is 14.67, whereas the mean number of neonates
produced in the 97.69^ effluent test sample population is 11.75.
Therefore, the difference in mean reproduction (DMR) is 2.92.
The Dunnett's test statistic can now be calculated as shown:
0.5
t = 2.92/[15.22*(1/12+1/12)]
t = 1.83
Because the purpose of the Dunnett's Test is only to detect a significant reduction in reproduction from the control, a one¬
sided test is appropriate. In this special case (i.e.,
comparison of a control group with only one sample group), the Dunnett's test is equivalent to the Student T Test. The critical value for the one-sided comparison, with a significance level of
0.01 (99X confidence level), 22 degrees of freedom, and 1 sample
population (excluding the control), is 2.51 (Appendix C, Table
C.4). The difference in reproduction is concluded to be non¬
significant because the calculated Dunnett's test statistic (1.83) is less than the critical value (2.51).
To quantify the sensitivity of the Dunnett's test, the
Minimum Significant Difference in Mean Reproduction (MSDMR) may
be calculated. The formula for the MSDMR (Horning and Weber, 1985) is as follows:
0.5
MSDMR = [CrT3*[MSE(1/n1+1/n2)]
where CrT = Critical test statistic in Table C.4 in Appendix C The absolute value of the difference between the MSDMR and
DMR (Difference in Mean Reproduction), multiplied by 12, will
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yield either the number of neonates in excess of the minimumnumber of neonates required to pass the test or the number of
neonates short of the minimum number of neonates to pass the
test. Continuing along with the example:
0.5
MSD = [2.51]*[15.22*(1/12+1/12)] = 4.00 and
abs[4.00 - 2.92]*12 = 13
Therefore, the reproduction data passed the Dunnett's test by 13
neonates.