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statistics/TeX_files/Experimental_design_and_ANOVA.tex index 1bdeb96..81ba13a 100644 --- a/Erasmus/Applied statistics/TeX_files/Experimental_design_and_ANOVA.tex +++ b/Erasmus/Applied statistics/TeX_files/Experimental_design_and_ANOVA.tex @@ -85,8 +85,109 @@ Power analysis allows us to determine the sample size required to detect an effe \subsection{Introduction to ANOVA} +Analysis of variance (ANOVA) is a statistical analysis for comparing means in experiments across different treatments. ANOVA is equivalent to analysing linear models. Before computers, ANOVA simplified calculations and it's still commonly used and referred to. + +Intuition for ANOVA: consider the variation within and between treatments. +\begin{center} + \begin{tikzpicture}[scale=0.9] + \begin{axis}[ + xmin=0.5, xmax=2.5, xlabel=treatment, + ymin=3, ymax=9, ylabel=some measure (some unit), + axis x line=bottom, + axis y line=left, + ] + \addplot[blue, only marks, mark=x] coordinates { + (1.00,5.27) + (1.00,5.92) + (1.00,3.87) + (1.00,5.43) + (1.00,5.16) + }; + \draw[cyan] (axis cs: 0.5,5) -- (axis cs: 1.5,5); + \addplot[red, only marks, mark=x] coordinates { + (2.00,6.73) + (2.00,7.61) + (2.00,7.15) + (2.00,6.99) + (2.00,7.14) + }; + \draw[orange] (axis cs: 1.5,7) -- (axis cs: 2.5,7); + \end{axis} + \end{tikzpicture} + \begin{tikzpicture}[scale=0.9] + \begin{axis}[ + xmin=0.5, xmax=2.5, xlabel=treatment, + ymin=3, ymax=9, ylabel=some measure (some unit), + axis x line=bottom, + axis y line=left, + ] + \addplot[blue, only marks, mark=x] coordinates { + (1.00,6.01) + (1.00,3.19) + (1.00,6.08) + (1.00,7.45) + (1.00,5.73) + }; + \draw[cyan] (axis cs: 0.5,5) -- (axis cs: 1.5,5); + \addplot[red, only marks, mark=x] coordinates { + (2.00,8.03) + (2.00,7.73) + (2.00,6.70) + (2.00,7.29) + (2.00,6.21) + }; + \draw[orange] (axis cs: 1.5,7) -- (axis cs: 2.5,7); + \end{axis} + \end{tikzpicture} +\end{center} + +In ANOVA, we consider +\begin{align} + F = \frac{\text{between-treatment variation}}{\text{within-treatment variation}} \notag +\end{align} + \subsubsection{One-way ANOVA} +Consider a one-factor completely randomised design, i.e. a number pf $p$ factor levels and experimental units assigned randomly to them. We have seen that this can be modelled as +\begin{align} + Y_i = \beta_0 + \beta_1x_{1i} + \dots + \beta_{p-1}x_{p-1,i} + \epsilon_i \notag +\end{align} +where the $x_{ji}$s are dummy variables for the factor levels. We wish to compare the means to response, $\mu_j$, across treatments $j$ and test the null hypothesis $H_0$: $\mu_1 = \mu_2 = \dots = \mu_p$. Suppose the model above, treatment 1 is the base level, them $\beta_0 = \mu_1$, $\beta_1 = \mu_2 - \mu_1$, ..., $\beta_{p-1} = \mu_p - \mu_1$. So the $H_0$ above is equivalent to $H_0$: $\beta_1 = \beta_2 = \dots = \beta_{p-1} = 0$. This is one-way ANOVA. It is the same as an F-test on the corresponding linear model. It shows that at least two treatment means differ. The F-statistic is computed from sums of squared errors (no model fitting required). + +\begin{example} + A study on the strength of different structural beams (\person{Hogg}, 1987). The MATLAB command is \texttt{anova1}. + + %TODO: Insert pic + table here + + ... this suggests that at least two beams differ in strength. +\end{example} + \subsubsection{Two-way ANOVA} -\subsection{Observational data - sampling} \ No newline at end of file +Consider a complete factorial design with two factors, one of which has three and the other has two levels. We have seen that this can be modelled as: +\begin{align} + Y_i = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i} + \beta_3w_{1i} + \beta_4x_{1i}w_{1i} + \beta_5x_{2i}w_{1i} + \epsilon_i \notag +\end{align} +where $x_{1i}$ and $x_{2i}$ are dummy variables for the first factor and $w_{1i}$ is a dummy variable for the second factor. Analogously to the one-way ANOVA, in two-way ANOVA, we perform a number of F-tests to compare the mean of the response across treatments. E.g. to test for interactions, we test $H_0$: $\beta_4 = \beta_5 = 0$. Testing if the mean response for levels of the first factor are equal, requires $H_0$: $\beta_1 = \beta_2 = 0$. Essentially, we use F-tests to compare nested models. These tests on multiple parameters simultaneously (e.g. if factors have more than 2 levels). They can show that at least two treatment means differ. + +\subsection{Observational data - sampling} + +Sometimes conducting experiments is not possible. Sampling methods are used to collect observational data in a systematic way. + +\begin{example} + Opinion poll to assess the voting intentions of the population before elections. It's not enough to just ask people in Bristol. +\end{example} + +Basic idea: consider a population. Ideally we would like to measure everyone. This is not possible. Sampling is the process of selecting a subset (a \begriff{statistical sample}) of units from the population to estimate whatever we are interested in for the whole population. +\begin{itemize} + \item \textbf{Probability sampling:} every unit in the population has a probability of being selected and this can be calculated. + \item \textbf{Nonprobability sampling:} not the product of a randomised selection process. +\end{itemize} + +Different sampling methods can be used, depending on information available, costs and accuracy requirements, e.g. +\begin{itemize} + \item \textbf{Simple random sampling:} all units in the population have the same probability of being selected (if the sample is small, this may not be representative). + \item \textbf{Systematic sampling:} arrange population in some order, select units at regular intervals. If starting point or order is randomised, this is a probability sampling. + \item \textbf{Stratified sampling:} organise population according to some categories into separate \begriff{strata} and sample randomly from those. + \item There are many additional methods, e.g. \textbf{voluntary sampling}, \textbf{accidental sampling}, \textbf{quota sampling},... +\end{itemize} \ No newline at end of file