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BioStat3425 Cheat Sheet (DRAFT) by

One-Way ANova Two-Way ANova Simple Linear regression Proportions

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Comparison Procedures

LSD Requires knowing the global F test is signif­icant before any testing should be done. More powerful than Bonfer­roni, but can lead to inflated Type I error rate. Not recomm­ended
TUKEY Controls experi­men­twise Type I error rate exactly if sample sizes between groups are equal More powerful than Bonfer­rroni Use when only interested in all (or most) pairwise compar­isons of means(­unp­lanned compar­isons)
BONFERRONI Very conser­vative Use when only interested in a small number of planned compar­isons Use when making compar­isons other than between pairs.

Individual Vs Family CI

**Indi­vidual confidence level (compa­ris­onw­ise­,𝛼𝑐𝑀): Success rate of a procedure for constr­ucting a single confidence interval (or conducting a single hypothesis test)
Familywise confidence level (exper­ime­ntw­ise­,𝛼𝑒𝑀):Success rate of a procedure for constr­ucting a family of confidence intervals (or a family of hypothesis tests) A β€œsucce­ssful” usage is one in which ALL intervals in the family capture their parameters

Pairwise Example

Nematodes in tomato plants How do nematodes affect plant growth? 12 identical pots, Different# of nematodes per pot; 0, 1000, 5000, Measure increase in tomato seedling height (cm) 16 days after planting
Hypothesis Test: Ho:u1=­u2=u3 vs Ha: Not all of the u are equal
Overall F test signif­icant (p-value = 0.0027) --> not all of the group means are equal
Hypo for pairwise compar­isons H0: ui-uj=0 Ha: ui-ujβ‰ 0 β†’ iβ‰ j
How many pairwise compar­isons are there for this example?
How many pairwise compar­isons are there for this example? K=K=I(­I-1)2β†’ 3(3-1)2= 3
Levels not connected by the same letter are signif­icantly different
Ordered Differ­ences Report for lower CL and upper Cl= difference of growth
Which nematode groups have signif­icantly different ang growth? 0-5000 (p -value = 0.0045)and 5000-1000 (0.006) are signif­icantly different Difference in growth between 0 and 5000 will be between 1.83 cm and 8.27 cm more than 5000

Factor Effects Model

π‘Œπ‘–π‘—π‘˜ = πœ‡+𝛼𝑖 +𝛽𝑗 +(𝛼𝛽)𝑖­π‘—+πœ€π‘–π‘—π‘˜
Response = overall avg growth + main effects +inter­action effect
Assump­tions: πœ€π‘–π‘—π‘˜~N­(0,𝜎2) are indepe­ndent Errors are normal, indepe­ndent, with constant variance
Constr­aints: σ𝛼𝑖 =σ𝛽𝑗 =Οƒ(𝛼𝛽)𝑖𝑗 =0 Required to keep model from being over-p­ara­met­erized
# of test = factor­alevels X factor­blevels

Assump­tions Errors are

Normal Probab­ility Plot
Constant Variance
Predicted Residual vs Fitted and Residual vs Factor Levels
Usually Assumed and typically assumed

Main Effects and Intera­ction Plot

Main Effects Plot Separate plot for each factor Plot of mean response for each level of the factor Gives an indication of whether a factor is important Horizontal line indicates means are the same for both levels of the factor and thus factor is not important
Intera­ction Plot Plot means for each treatment combin­ation against levels of one factor, with different lines for the other factor Parallel Lines – No Intera­ction Crossing Lines - Intera­ction

Simple Liner Regression Vocal

Least Squares Regression
standard approach for estimating the line
The line is chosen, so that the sum of squared vertical distances between points and the line is minimized.
correl­ation coeffi­cient r
A measure of associ­ation for two quanti­tative variables is the
predicted or fitted values.
The points that lie on the regression line vertically above (or below) an observed value Notation ^yi
(vertical) distances between observed and predicted values notatio: ei=yi-^yi



First: Test for Intera­ction β€’ 𝐻0: (𝛼𝛽)𝑖𝑗=0 for all i,j vs. π»π‘Ž: π‘π‘œπ‘‘ π‘Žπ‘™π‘™ (𝛼𝛽)𝑖𝑗 equal zero
p-value <a =Reject HO or p-valu­e>a = fail to reject H0 This means that both factors are important (even if main effects not signif­icant) β€’ We may need to interpret in terms of the intera­ction since we may not be able to separate out the main effects for individual interp­ret­ation
Then: Test Main Effects Factor A 𝐻0:𝛼𝑖 =0 for all i vs. π»π‘Ž:π‘π‘œπ‘‘­π‘Žπ‘™π‘™π›Όπ‘– equal zero β€’ Main Effect of Factor B 𝐻0:𝛽𝑗 =0 for all j vs. π»π‘Ž:π‘π‘œπ‘‘­π‘Žπ‘™π‘™­π›½π‘—equal zero
H0: factorAi= 0 for all i (no effect) reject H0 if p-valu­e<a

Model parame­ter­s(u­nknown)


Linear relati­onship between X and Y
Model assumes that the error terms are
Indepe­ndent, Normal, Have constant variance.
Residuals may be used to explore the legitimacy of these assump­tions.