Lecture 6 Comparison Between Treatment Means PDF

06/11/2022 Lecture 6. Comparison between Treatment Means AGRI 195 Research Methods in Agriculture and Food Science...

06/11/2022 Lecture 6. Comparison between Treatment Means AGRI 195 Research Methods in Agriculture and Food Science 1st Semester A.Y. 2022-2023 1 Pair comparisons Used to compare specific treatments to each other this is referred as pre-planned comparison Unplanned comparison – no specific comparison is chosen. Every possible pair of treatment means is compared. The probability of committing Type I error is greater, when more comparisons are made in an experiment Type I error is declaring a significant difference, when in fact there is none. (False rejection of Ho) AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 2 2 1  06/11/2022 Least Significant Difference (LSD) Appropriate for planned comparisons Experiment-wise error rate increases with increase in no. of pairs of treatments compared. Advisable to apply only when treatment effect from F-test is significant. AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 3 3 F-protected LSD Performed and used only when the Ho is rejected LSD can be computed even if the researcher failed to reject Ho. However, this might wrongly suggest that there are significant differences between treatment means. Therefore, LSD should only be calculated when Ho is rejected. AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 4 4 2  06/11/2022 If the difference between two treatment means is greater than the LSD, then the two treatment means are significantly different from each other at α level of significance. AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 5 5 AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 6 Gomez and Gomez, 1984 6 3  06/11/2022 Split-plot arrangement AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 7 7 Strip-plot arrangement AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 8 8 4  06/11/2022 Group Comparisons Each comparison was tested by calculating a single degree of freedom linear contrast compares the mean of one group vs. the mean of another group Contrasts must always be pre-planned! Linear contrast involves partitioning the df and SS of the treatment If the comparisons are independent, they are called orthogonal. AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 9 9 Advantages 1. Enable to answer specific, important questions about treatment effects 2. Simple computations 3. Provide useful check on treatment SS AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 10 10 5  06/11/2022 Orthogonal coefficients Comparison for coefficients are constructed using the following rules 1. If two groups of equal size are to be compared, assign coefficient of +1 to the members of one group and -1 to those of the other group 2. In comparing groups containing different numbers of treatments, assign to the first group, coefficients equal to the number of treatments in the second group, and to the second group, coefficients of the opposite sign equal to the number of treatments in the first group AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 11 11 Orthogonal coefficients 3. Reduce coefficients to the smallest possible integers. 4. Interaction coefficients can always be found by multiplying the corresponding coefficients of the main effects A1B1 A1B2 A2B1 A2B2 A1 vs. A2 B1 vs. B2 AxB AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 12 12 6  06/11/2022 For comparisons to be orthogonal: 1. The sum of the coefficients must equal to zero 2. The sum of the product of the corresponding coefficients of any two comparisons is zero AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 13 13 We calculate the contrast test by using the formula AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 14 14 7  06/11/2022 Example Experiment: No. of corn seedlings surviving a stalk rot infection Treatments: A= untreated check E= non-Hg fungicide #2 B= Hg fungicide #1 F= non-Hg fungicide #3 C= Hg fungicide #2 G= non-Hg fungicide #4 D= non-Hg fungicide #1 H= non-Hg fungicide #5 AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 15 15 We can compare Hg fungicides to non-Hg fungicides. Ho: μHg=μnon-Hg Ha: μHg≠μnon-Hg Need to compare treatments B and C with D,E,F,G, and H So, AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 16 16 8  06/11/2022 If Q is very small or near zero, then we don’t reject the Ho, and we assume no difference between the two groups. However, if Q is large, then it’s possible that the two groups are different, then we test the significance of the formula by calculating AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 17 17 Contrast df = 1 AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 18 18 9  06/11/2022 Any two contrasts are orthogonal (independent), if the product of their coefficients summed up to zero Independent contrasts imply non-redundancy of information Mutual orthogonality is desirable but not absolutely essential If the scientist is interested on several contrasts, he/she should not let the lack of orthogonality prevent him from performing the statistical tests AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 19 19 Trend Comparison Individual df comparisons for equally spaced treatments (incremental) Opportunity to look at the response curve of the data -- multiple regression Similar analysis can be done using regression analysis however, this is simpler! AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 20 20 10  06/11/2022 Partition the factor in the ANOVA table into separate single df comparisons, like in linear contrasts. No. of possible comparisons =no. of levels of a factor minus one i.e. 5 levels of N, therefore 4 possible comparisons These comparisons are called orthogonal contrasts or comparisons, also called trend comparisons. AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 21 21 Orthogonal polynomials are equations such that each is associated with a power of the independent variable (or factor), i.e. X-linear, X2-quadratic, X3-cubic, etc. 1st order comparisons measure linear relationships 2nd order comparisons measures quadratic relationships 3rd order comparisons measures cubic relationships AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 22 22 11  06/11/2022 ANOVA RCBD RCBD with polynomial contrasts Source of variation df Source of variation df Rep r-1 Rep r-1 Treatment t-1 Treatment t-1 Error (r-1)(t-1) Linear 1 Total rt-1 Quadratic 1 Error (r-1)(t-1) Total rt-1 AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 23 23 Sum of squares can be calculated for each comparisons using the formula: F-tests can be calculated for each of the polynomial contrasts. AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 24 24 12  06/11/2022 The coefficients used for each polynomial contrasts AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 25 25 Effect of row spacing on yield (bu/ac) of soybean Given the following row spacing: 18 inches 24 30 36 42 What will you expect with the yield of soybean at wider row spacing? What will you expect with the yield of soybean at shorter row spacing? AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 26 26 13  06/11/2022 Effect of row spacing on yield (bu/ac) of soybean AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 27 27 Step 1. ANOVA of data as an RCBD Why do we need to perform ANOVA??? AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 28 28 14  06/11/2022 Step 2. Partition Treatment (SOV) into 4 single df orthogonal polynomial contrasts AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 29 29 The coefficients used for each polynomial contrasts AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 30 30 15  06/11/2022 Step 3. Calculate Sum of Squares for each contrast AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 31 31 Step 4. Rewrite ANOVA AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 32 32 16  06/11/2022 Linear vs. Quadratic 37.00 35.00 33.00 31.00 Linear (Series1) Poly. (Series1) 29.00 27.00 25.00 0 10 20 30 40 50 AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 33 33 Step 5. Conclusion Linear and quadratic effects for row spacing treatments are highly significant. The linear component is the portion of the SS attributable to the linear regression of yield on row spacing. The quadratic component measures the additional improvement due to fitting the X2 component AGRI 195: Research Methods in Agriculture and Food Science | 1st sem AY 2022-2023 | Lecture 6 34 34 17  06/11/2022 End of Lecture 35 18