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Biometry 242 – NS 12 -14
NS 12 – Split Plot Design * Factor A on large plots (Main plots), Factor B on smaller plots (Sub-plots)

* Use a split-plot when: * there is a limited number of plots * one of the factors is more important than the other

* More NB Factor on sub-plots * Disadvantage – one factor forfeits precision * Has two error terms

= SSMAIN
= SSTMT
= SSMAIN
= SSTMT

* RSSMAIN and RSSTMT → Calculate but don’t use in ANOVA * Also need RSSBLOCKS, RSSPP, and RSSM (CT and RSSTOTAL) * F values are calculated using the error from the same block * For t-test * Standard errors: * Error (b)n for interaction 9.78583 * 2 × Error (b)n for Factor M 2 × 9.78586 * 2 × Error (a)n for Factor PP 2 × 4.96756 * 2 critical-t values → t at 2 and t at 4 df i.e. 4.303 and 2.776

* Could ask: do ANOVA and t-test, or ANOVA and interpret results from F; Standard error for the difference (a or b); Conclusion: levels differ/do not differ at 1% etc.

NS 13 – Non-parametric tests

* Parametric tests for data with normal distribution (t, F or X2 distribution)

* Non-parametric tests for * Categorical data, * Quantitative data divided into class intervals, * Small data sets, * Data sets without repetition of the TMTs.

* Non- parametric tests * Medians, not Means * Usually rank your data

* Single sample: * Sign test (No assumptions about distribution) * Rank test (assumes data comes from symmetrical distribution) * Wilcoxon’s symmetry test

* For 2 independent samples * Mann-Whitney U test (Assumes distributions have same shape and equal variances)

* For small experiments * For CR design → Kruskal-Wallis test * For RB design → Friedman test

* Sign test * Equivalent to t-test for single mean for normal data * For small data sets obtained from a uniformity trial * Data has skewed distribution * H0 : Median is equal to M * If true, half measurements will be smaller than M and other half bigger. * Ratio of - : + should be 1:1 * Compare proportion of +’s and –‘s to binomial distribution

* Mann-Whitney U test * equivalent to pair-wise t-test for normal data * Used for small data sets from CR experiments with 2 TMTs * H0 : Different groups have the same medians * Steps: * Construct array * Give ranks * Use formula for each group (TMT) * Ui= Ri- ni(ni+1)2
(R = rank total for TMT i) (n = number of measurements in group i) * Smaller Ui is the test measure

* Kruskall-Wallis test * Equivalent to one-way ANOVA for normal data * Used for small data sets, from CR experiments with 2 or more TMTs * H0 : Different groups have the same medians * Steps: * Construct array * Give ranks * Test measure calculated using:
K= 12N(N+1)Ri2ni-3(N+1) * N = total number of measurements * ni = number of measurements in group i * Ri = Rank total for TMT i * Find p-value from χ2 table at (no. groups minus 1)df * p < 0.05 reject (so at least one median differs)

* Friedman test * Equivalent to two-way ANOVA for normal data * Used for small data sets, from RB experiments with 2 or more TMTs * H0 : Different groups have the same medians * Most NB difference from KW test * separate array is constructed for each block * then ranks awarded within each block * Steps: * Construct separate arrays * Give ranks within blocks * Test measure calculated using:
S=12N(g+1)Ri2-3b(g+1) (g = no. of groups)(b = no. blocks) * Find p-value from χ2 table at (no. groups minus 1)df * p < 0.05 reject (so at least one median differs)

* Exercises pg 147
NS 14 – Multiple Linear Regression * Multiple linear regression model: Y=B0+B1X1+B2X2+…+∈ *

* Be able to write regression function i.e. Y = - 11.6051 + 0.0001 X1 + 0.0018X2 + 0.4223 X3

* Interpret R2 value (coefficient of determination) * = 0.6256 * So 62.56% of the variation of Y can be explained by the model. * Higher R2 means model fits the data better

* F-ratio tests: * H0: All partial slops are = 0 (B1 = B2 = … = 0) * Ha: At least one of the partail slopes is not = 0 * * T-test to see which variables should be included in the model * E.g. H0: B1 = 0 * E.g. Ha: B1≠ 0 * Know where to get t-critical * If reject H0 then that variable should be included * Say which variables contribute * Understand: X2 has a positive effect on Y and for each unit increase in X2, the yield will increase by 0.0018 units.

* Correlation coefficient = r =R2 (correlation between actual and predicted values)

* Better measure to use (than R2) * Is adjusted R2 (Adjusted coefficient of determination) * The larger the adjusted R2, the better the fit of the model * Use to find best model

* Assumptions for mutiple linear regession model: * Homoscedastic * Normal * Independent

* Test for homoscedasticity * Residual plot * Residuals spread evenly = homoscedastic * Not homoscedastic = pattern, funnel shape * U shape usually means quadratic relationship * Can show outliers (point far away from zero line) * Outlier is when residual is > 2 times the residual standard error * Look at output

* Test for normality * Normal Q-Q plot * Residuals from smallest to largest * Normal if values form a straight diagonal line

* Increase accuracy by increasing number of observations!!

* Making predictions * Substitute values into X’s * And calculate Y

Formulas Standard errors: | | for interaction | Error (b)n e.g. 9.78583 | for Main Factor | 2 × Error (a)n e.g. 2 × 4.96756 | for Sub Factor | 2 × Error (b)n e.g. 2 × 9.78586 | Mann-Whitney U test | Ui= Ri- ni(ni+1)2(R = rank total for TMT i) (n = number of measurements in group i) | Kruskall-Wallis test | K= 12N(N+1)Ri2ni-3(N+1)N = total number of measurementsni = number of measurements in group iRi = Rank total for TMT i | Friedman test | S=12N(g+1)Ri2-3b(g+1)(g = no. of groups)(b = no. blocks) |