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% Document Info
\author{Hsuzc}
\pdfinfo{
  /Title (biostat3425.pdf)
  /Creator (Cheatography)
  /Author (Hsuzc)
  /Subject (BioStat3425 Cheat Sheet)
}

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\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{BioStat3425 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Hsuzc} via \textcolor{DarkBackground}{\uline{cheatography.com/171419/cs/35971/}}}
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\noindent
\begin{multicols}{3}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Hsuzc \\
  \uline{cheatography.com/hsuzc} \\
  \end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 9th December, 2022.\\
   Page {\thepage} of \pageref{LastPage}.
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\end{multicols}}




\begin{document}
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Comparison Procedures}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{{\bf{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} \tn 
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% Row 1
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\mymulticolumn{1}{x{5.377cm}}{{\bf{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)} \tn 
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% Row 2
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\mymulticolumn{1}{x{5.377cm}}{{\bf{BONFERRONI}} Very conservative Use when only interested in a small number of planned comparisons Use when making comparisons other than between pairs.} \tn 
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Individual Vs Family CI}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{**Individual confidence level (comparisonwise,𝛼𝑐𝑤): Success rate of a procedure for constructing a single confidence interval (or conducting a single hypothesis test)} \tn 
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% Row 1
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\mymulticolumn{1}{x{5.377cm}}{{\bf{Familywise confidence level (experimentwise,𝛼𝑒𝑤):}}Success rate of a procedure for constructing a family of confidence intervals (or a family of hypothesis tests) A "successful" usage is one in which ALL intervals in the family capture their parameters} \tn 
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\end{tabularx}
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Pairwise Example}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{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} \tn 
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% Row 1
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\mymulticolumn{1}{x{5.377cm}}{Hypothesis Test: Ho:u1=u2=u3 vs Ha: Not all of the u are equal} \tn 
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\mymulticolumn{1}{x{5.377cm}}{Overall F test significant (p-value = 0.0027) -{}-\textgreater{} not all of the group means are equal} \tn 
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\mymulticolumn{1}{x{5.377cm}}{Hypo for pairwise comparisons  H0: ui-uj=0 Ha: ui-uj≠0 → i≠j} \tn 
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\mymulticolumn{1}{x{5.377cm}}{How many pairwise comparisons are there for this example?} \tn 
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\mymulticolumn{1}{x{5.377cm}}{How many pairwise comparisons are there for this example? K=K=I(I-1)2→ 3(3-1)2= 3} \tn 
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\mymulticolumn{1}{x{5.377cm}}{Levels not connected by the same letter are significantly different} \tn 
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\mymulticolumn{1}{x{5.377cm}}{Ordered Differences Report for lower CL and upper Cl= difference of growth} \tn 
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\mymulticolumn{1}{x{5.377cm}}{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} \tn 
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Factor Effects Model}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{𝑌𝑖𝑗𝑘 = 𝜇+𝛼𝑖 +𝛽𝑗 \seqsplit{+(𝛼𝛽)𝑖𝑗+𝜀𝑖𝑗𝑘}} \tn 
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\mymulticolumn{1}{x{5.377cm}}{Response = overall avg growth + main effects +interaction effect} \tn 
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\mymulticolumn{1}{x{5.377cm}}{Assumptions: 𝜀𝑖𝑗𝑘\textasciitilde{}N(0,𝜎2) are independent  Errors are normal, independent, with constant variance} \tn 
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% Row 3
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\mymulticolumn{1}{x{5.377cm}}{Constraints:  σ𝛼𝑖 =σ𝛽𝑗 =σ(𝛼𝛽)𝑖𝑗 =0 Required to keep model from being over-parameterized} \tn 
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\mymulticolumn{1}{x{5.377cm}}{\# of test = factoralevels X factorblevels} \tn 
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\begin{tabularx}{5.377cm}{x{1.4931 cm} x{3.4839 cm} }
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Assumptions Errors are}}  \tn
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Normal & Normal Probability Plot \tn 
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Constant Variance & Predicted Residual vs Fitted and Residual vs Factor Levels \tn 
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\seqsplit{Independence} & Usually Assumed and typically assumed \tn 
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Main Effects and Interaction Plot}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{{\bf{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} \tn 
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% Row 1
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\mymulticolumn{1}{x{5.377cm}}{{\bf{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} \tn 
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\begin{tabularx}{5.377cm}{x{1.9908 cm} x{2.9862 cm} }
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Simple Liner Regression Vocal}}  \tn
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Least Squares  Regression & standard approach for estimating the line\{\{nl\}\}The line is chosen, so that the sum of squared vertical distances  between points and the line is  minimized. \tn 
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correlation coefficient r & A measure of association for two quantitative variables is the \tn 
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predicted or fitted values. & The points that lie on the regression line vertically above (or below)  an observed value Notation \textasciicircum{}yi \tn 
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residuals. & (vertical) distances between observed and predicted values  notatio: ei=yi-\textasciicircum{}yi \tn 
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Model}}  \tn
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/hsuzc_1670555320_Screen Shot 2022-12-08 at 9.08.14 PM.png}}} \tn 
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{F-Tests}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{First: Test for Interaction • 𝐻0: (𝛼𝛽)𝑖𝑗=0 for all i,j vs. 𝐻𝑎: 𝑁𝑜𝑡 𝑎𝑙𝑙 (𝛼𝛽)𝑖𝑗 equal zero} \tn 
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% Row 1
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\mymulticolumn{1}{x{5.377cm}}{p-value \textless{}a =Reject HO or p-value\textgreater{}a = fail to reject H0 This means that both factors are important (even if main effects not  significant)  • 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} \tn 
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% Row 2
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\mymulticolumn{1}{x{5.377cm}}{Then:  Test Main Effects Factor A 𝐻0:𝛼𝑖 =0 for all i    vs. \seqsplit{𝐻𝑎:𝑁𝑜𝑡𝑎𝑙𝑙𝛼𝑖} equal zero • Main Effect of Factor B 𝐻0:𝛽𝑗 =0 for all j    vs. \seqsplit{𝐻𝑎:𝑁𝑜𝑡𝑎𝑙𝑙𝛽𝑗equal} zero} \tn 
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% Row 3
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\mymulticolumn{1}{x{5.377cm}}{H0: factorAi= 0 for all i (no effect) reject H0 if p-value\textless{}a} \tn 
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\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Model parameters(unknown)}}  \tn
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/hsuzc_1670555370_Screen Shot 2022-12-08 at 9.09.02 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\begin{tabularx}{5.377cm}{x{2.43873 cm} x{2.53827 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Assumprions}}  \tn
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\mymulticolumn{2}{x{5.377cm}}{Linear relationship between X and Y} \tn 
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Model assumes that the error terms are & Independent, Normal, Have constant variance. \tn 
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\mymulticolumn{2}{x{5.377cm}}{Residuals may be used to explore the legitimacy of these  assumptions.} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}--}
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% That's all folks
\end{multicols*}

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