Least significant difference

Once a significant difference has been demonstrated by an analysis of variance, a modified version of the f-test, known as Fisher s least significant difference, can be used to determine which analyst or analysts are responsible for the difference. The test statistic for comparing the mean values Xj and X2 is the f-test described in Chapter 4, except that Spool is replaced by the square root of the within-sample variance obtained from an analysis of variance. 

Individual comparisons using Fisher s least significant difference test are based on the following null hypothesis and one-tailed alternative hypothesis... 

Fisher s least significant difference a modified form of the f-test for comparing several sets of data. (p. 696) flame ionization detector a nearly universal GC detector in which the solutes are combusted in an H2/air flame, producing a measurable current, (p. 570)... 

Fig. 16.42 Atrazine degradation in sterilized aquifer material (Al, Bl, Cl) and in natural (unamended) aquifer material (A2, B2, C2) for reactors under stagnant (15°C), stagnant (25°C), and recirculating (15°C) conditions. Data points are the mean of three replications. The least significant difference between these treatments at the p < 0.05 level is 0.075. Figures are identified as (A) Topeka, (B) Ashland, and (C) Hutchinson aquifers. (Schwab et al. 2006)...

Statistical Analysis. Statistical analyses (two-way ANOVA) were performed by using the Statistical Analysis System (SAS, 1990). Means were compared by the least significant difference (LSD) test at a = 0.05. 

Significantly different (p < 0.05) by one-way analysis of variance (ANOVA) with the Least Significant Difference post hoc test using JMP program from SAS, Cary, NC on a Macintosh. Reproduced with permission from (Ischiropoulos et al., 1992a). 

Sometimes we may be interested in knowing how large a difference would be needed in order to reach the desired level of significance. Here concept of least significant difference or LSD may be of interest. 

Least significant difference, 22 Least squares line, 36,37 Linear programming, 65... 

Average values followed by the same letter were not significantly different at the 0.05 level using Least Significant Difference at P=0.05. 

Foliar disease scored on scale of 0 - 5, where 0 = no disease and 5 = plant defoliated due to foliar disease. 5Average values followed by the same letter were not significantly different at the 0.05 level using a Least Significant Difference. 

Average of three replications. Values with the same superscript letter in a given column indicate no significant difference (p < 0.05). SEM, standard error of means LSD, least significant difference. 

The difference between adjacent values clearly shows that there are no significance differences in the means, as the least significant difference, 9.63, is much larger than any of the differences between the pairs of results (the largest difference is between A and C is only 1.0 in magnitude). 

NS = not significant LSDto.ooi = least significant difference (standard error of difference x Student s test value) at the 0.001 level = significant at the 0.001 level... 

Figure 1. Sensory descriptive analysis data of Napa Cabernet Sauvignon samples and the base wine. Mean ratings of 14 judges x 2 replicates and least significant differences (LSD, p<0.05) are shown. For sample codes, see Table II.

With a 1.7 correction factor (Saka et al. Least significant difference at p = 0.0.5. 

For data satisfying the ANOVA requirements, the least significant difference (LSD) is useful for making planned comparisons among several means. Any two means that differ by more than the LSD will be significantly different. The LSD is useful for showing on graphs. 

This analysis allows us to split the variability observed for B into contributions due to different factors. The probability (p-value) provides a measure of the statistical significance (at a confidence level of 95%) of each factor. Overall at least one of the factors has had a significant effect (p = 0.0001) on the measured level of B. This is in a good agreement with previous observations. Multiple range tests (Fisher s least significant difference (LSD)) was performed to determine which of the treatment means were significantly different from each other, and the results are summarized in Table 4.5.5. 

If significant differences are indicated in ANOVA, we are often interested in the cause. Is one mean different from the others Are all the means different Are there two distinct groups that the means fall into There are several methods to determine which means are significantly different. One of the simplest is the least significant difference (LSD) method. In this method, a difference is calculated that is judged to be the smallest difference that is significant. The difference between each pair of means is then compared with the least significant difference to determine which means are different. 

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