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\begin{center}   
{\Large \textbf{STA 305s14 Regular Assignment Three}}\footnote{Copyright information is at the end of the last page.}
\vspace{1 mm}
\end{center}

\noindent
This assignment is preparation for Term Test Two on March 10th, and for the final exam. Your solutions to these homework problems will not be handed in. Use the formula sheet, which is posted on the course home page. As more material is covered, additional problems will be added at the end of the assignment.

% \vspace{3mm}

\begin{enumerate}

%%%%%%%%%% 

\section*{Lecture Unit 6: Analysis of variance methods for a one-factor completely randomized design}

\item \label{backpain} In a study of remedies for lower back pain, volunteer patients at a back clinic were randomly assigned to one of seven treatment conditions:
    \begin{itemize} 
        \item OxyContin: A pain pill in the opiate family.
        \item Ibuprofen: A non-steroidal anti-inflammatory drug (Advil, Motrin)
        \item Acupuncture: The insertion and manipulation of thin needles into specific points on the body to relieve pain or for therapeutic purposes.
        \item Chiropractic: A form of therapy that includes manipulation of the spine, other joints and soft tissue.
        \item Stress reduction training based on thinking positive thoughts, a treatment that theoretically should not be effective. This is the non-drug control condition.
        \item Placebo: A sugar pill; patients were told that it was a pain killer with few side effects. This is the drug control condition.
        \item Waiting list control: Patients were told that the clinic was overcrowded (true), and that they would were on a waiting list. This group received no treatment at all, not even a pretend treatment --- until the study was over, at which point they received the most effective treatment based on the results of the study. 
    \end{itemize}

Degree of reported pain was measured by a questionnaire before treatment began, and again after six weeks. The dependent variable was Before-minus-After difference in reported pain, which will be called ``improvement," or ``effectiveness."

The idea is that the effectiveness of the drug treatments should be assessed relative to the drug control (placebo), while the effectiveness of the non-drug treatments should be assessed relative to the non-drug control (stress reduction training). Improvement in the control conditions can be measured relative to no treatment at all.

    \begin{enumerate} 
        \item You will use a regression model with an intercept and indicator (zero-one) dummy variables. Make a table showing how you would set up the dummy variables. There is more than one reasonable way to do this.
        \item Add another column to the end of your table, showing the expected improvement in back pain in terms of your $\beta$ parameters.
        \item For each of the questions below, give the null hypothesis in terms of $\beta$ parameters. This is a scientific study, and the results will not be ignored if they are the opposite of what's predicted. So even when the question seems to imply a directional alternative, all the tests are non-directional, and the null hypothesis says that something is \emph{equal} to something else.
        \begin{enumerate}
            \item Does OxyContin work any better than the placebo?
            \item Does Ibuprofen work any better than the placebo?
            \item Do Chiropractic treatment and Stress reduction training differ in their effectiveness?
            \item Which results in more mean improvement, Acupuncture or Stress reduction training?
            \item Is the average improvement from the two drug therapies different from the improvement from the placebo?
            \item Does either drug therapy differ from the placebo in its effectiveness? (This is a single test of two equalities.)
            \item Does either non-drug therapy differ in effectiveness from Stress reduction training?
            \item Is the Placebo better than no treatment at all?
            \item Is Stress reduction training better than no treatment at all?
            \item s the average effectiveness of the drug therapies different from the average effectiveness of the non-drug therapies?
            \item Do Stress reduction training and the Placebo differ in their effectiveness?
            \item Does either control condition (Drug or Non-Drug) differ from no treatment at all?
            \item Is treatment condition (the full independent variable, including the No Treatment condition) related to improvement?
        \end{enumerate}
    \end{enumerate}

\item For the random sampling model (not the randomization model) explain how the assumption of unit-treatment additivity implies equal variances. Use the example of an experiment with a control condition and just one experimental treatment.

\item \label{onesample} Let $Y_1, \ldots, Y_n$ be a random sample (i.i.d.) from a $N(\mu,\sigma^2)$ distribution. 
    \begin{enumerate}
        \item Write this as a regression model in matrix form. 
            \begin{enumerate}
                \item What is $\mathbf{Y}$? What are its dimensions?
                \item What is $\mathbf{X}$? What are its dimensions?
                \item What is $\boldsymbol{\beta}$? What are its dimensions?
                \item What is $\boldsymbol{\epsilon}$? What is its distribution?
                \item What is $\mathbf{X}^\prime\mathbf{X}$?
                \item What is $(\mathbf{X}^\prime\mathbf{X})^{-1}$?
                \item What is $\mathbf{X}^\prime\mathbf{Y}$?
                \item What is $\widehat{\boldsymbol{\beta}}$?
                \item What is the $n \times 1$ vector $\widehat{\mathbf{Y}}$?
                \item What is $SSE$?
            \end{enumerate}
        \item Cite the fact from the formula sheet that tells you $\sum_{i=1}^n(Y_i-\overline{Y})^2$ and $\overline{Y}$ are independent.
        \item Cite the fact from the formula sheet that tells you $\frac{\sum_{i=1}^n(Y_i-\overline{Y})^2}{\sigma^2} \sim \chi^2(n-1)$.
    \end{enumerate}



\item Consider once again the experiment on scab disease in potatoes, which first appeared in Computer Assignment 2. Remember that there were three levels of sulphur (300, 600 and 1200 pounds per acre) and a control.  
    \begin{enumerate} 
        \item Make a table showing how you would set up the dummy variables for \emph{cell means} dummy variable coding. That's the one with indicators and no intercept. Add another column showing the expected value of the response for each experimental condition, including the control. 
        \item Write this as a regression model in matrix form. 
            \begin{enumerate}
                \item What is $\mathbf{Y}$? What are its dimensions?
                \item What is $\mathbf{X}$? What are its dimensions? 
                \item What is $\boldsymbol{\beta}$? What are its dimensions?
                \item What is $\boldsymbol{\epsilon}$? What is its distribution?
                \item What is $\mathbf{X}^\prime\mathbf{X}$? This is easier to see if data from the same experimental condition are in adjacent rows.
                \item What is $(\mathbf{X}^\prime\mathbf{X})^{-1}$?
                \item What is $\mathbf{X}^\prime\mathbf{Y}$?
                \item What is $\widehat{\boldsymbol{\beta}}$?
                \item What is the \emph{distribution} of $\widehat{\boldsymbol{\beta}}$? Give the $4 \times 4$ covariance matrix explicitly. 
                \item What is $\widehat{\mathbf{Y}}$?
            \end{enumerate}
        \item Now change notation, letting $Y_{ij} = \mu_j + \epsilon_{ij}$, for $j=1, \ldots, p$ and $i = i, \ldots, n_j$. For the scab disease example, what is $p$? Now in general,
            \begin{enumerate}
                \item What is the joint distribution of the $\epsilon_{ij}$? 
                \item In this new notation, what is $\overline{Y}_j$?
                \item Let $\overline{Y}$ denote the sample mean of all the observations. White a formula for $\overline{Y}$ in terms of the new notation.
                \item In the new notation, what is $SST$?
                \item In the new notation, what is $SSE$?
                \item In the new notation, what is $SSR$?
                \item What is the distribution of $\frac{SSE}{\sigma^2}$?
                \item Under $H_0: \mu_1 = \cdots = \mu_p$, what is the distribution of $\frac{SST}{\sigma^2}$? Why can you just use question~\ref{onesample}?
                \item Write $\overline{Y}$ as an explicit function of the $\overline{Y}_j$.
                \item How do you know $SSR$ and $SSR$ are independent for this model?
                \item You will notice that the formula sheet now has this, which was proved in STA302: If  $W=W_1+W_2$ with $W_1$ and $W_2$ independent, $W\sim\chi^2(\nu_1+\nu_2)$, $W_2\sim\chi^2(\nu_2)$ then $W_1\sim\chi^2(\nu_1)$. Use this fact to find the distribution of $\frac{SSR}{\sigma^2}$. Why does your conclusion depend on $H_0$ being true?
                \item Based on the newly revised formula sheet, give the formula for a test statistic for testing $H_0: \mu_1 = \cdots = \mu_p$. What is its distribution when $H_0$ is true? Why might you expect big values when $H_0$ is false? It's also the test statistic you'd get if you carried out a general linear test, but that's not obvious. Don't just give the formula for the general linear test. 
            \end{enumerate}
    \end{enumerate}

\item Suppose a completely randomized design is used to compare the expected response for five experimental treatments. 
    \begin{enumerate}
        \item Write a regression model with an intercept for this problem.
        \item Make a table showing how you would set up dummy variables with effect coding. That's the scheme with the minus ones.
        \item Define the ``grand mean" by $\mu = \frac{1}{p}\sum_{j=1}^p\mu_j$. For this problem (with $p=5$), what is $\mu$ in terms of the $\beta$ values? Show your work.
        \item In terms of the $\beta$ values, what is $\tau_3 = \mu_3-\mu$? Show a little work.
        \item In terms of the $\beta$ values, what is $\tau_5 = \mu_5-\mu$? Show a little work.
        \item Write $H_0: \mu2=\mu_3$ in terms of $\beta$ values. 
        \item Write $H_0: \mu4=\mu_5$ in terms of $\beta$ values. Simplify a bit.
    \end{enumerate}

\pagebreak

% contrasts, mostly.
\item  In this question, ``test a contrast" is a short way to say test the null hypothesis that a contrast of the $\mu_j$ values is equal to zero. The ``weights" of a contrast are the $a_j$  constants in $c = a_1\mu_1  + \cdots + a_p\mu_p$.   Here's the setting. Three hundred university student volunteers who wanted to lose weight were weighed in a clinic under controlled conditions. Then they were randomly assigned to one of six treatment groups:
\begin{itemize}
     \item[1] Free Health Club membership without personal trainer
     \item[2] Free Health Club membership with personal trainer: Emphasis on aerobic conditioning
     \item[3] Free Health Club membership with personal trainer: Emphasis on strength training
     \item[4] Free vegetarian cooking and diet class
     \item[5] Free Exercise video
     \item[6] Waiting list control (They were told ``Sorry, we'll call you when there's an opening.")
\end{itemize}
After six months they were weighed again. The dependent variable is weight loss in kilograms: Before minus After. 

\begin{enumerate}
     \item Is average weight loss in the exercise video condition more than average weight loss for the waiting list control condition?
        \begin{enumerate}
            \item State the null hypothesis in terms of $\mu_j$ values.
            
%           ($\mu_5=\mu_6$) 
            \vspace{5mm}
            \item In the table below, give the weights of the contrast or contrasts you would test to answer the question. There should be one row for each contrast.          
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}  \hline
      ~1~  &   ~2~ & ~3~ &   ~4~ &  ~5~ &   ~6~ \\ \hline
           &       &     &       &      &        \\ \hline
\end{tabular}
\end{center}
%\begin{center}
%\begin{tabular}{|c|c|c|c|c|c|}  \hline
%      ~1~  &   ~2~ & ~3~ &   ~4~ &  ~5~ &   ~6~ \\ \hline
%       0   &    0  &  0  &    0  &   1  &    -1   \\ \hline
%\end{tabular}
%\end{center}

        \end{enumerate}
     \item Is average weight loss different for the three treatments that include a health club membership?
        \begin{enumerate}
            \item State the null hypothesis in terms of $\mu_j$ values.  % ($\mu_1=\mu_2=\mu_3$)
            \vspace{5mm}
            \item In the table below, give the weights of the contrast or contrasts you would test to answer the question. There should be one row for each contrast.          
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}  \hline
      ~1~  &   ~2~ & ~3~ &   ~4~ &  ~5~ &   ~6~ \\ \hline
           &       &     &       &      &        \\ 
           &       &     &       &      &        \\ 
           &       &     &       &      &        \\ 
           &       &     &       &      &        \\ \hline
\end{tabular}
\end{center}
%\begin{center}
%\begin{tabular}{|c|c|c|c|c|c|}  \hline
%      ~1~  &   ~2~ & ~3~ &   ~4~ &  ~5~ &   ~6~ \\ \hline
%       1   &   -1  &  0  &    0  &   0  &    0   \\  \hline
%       0   &    1  & -1  &    0  &   0  &    0   \\  \hline
%\end{tabular}
%\end{center}

        \end{enumerate}

     \item Consider a test for differences among the three treatments that include a health club membership, and \emph{at the same time}, for the three treatments that do not include a health club membership

        \begin{enumerate}
            \item State the null hypothesis in terms of $\mu_j$ values. % ($\mu_1=\mu_2=\mu_3$ and $\mu_4=\mu_5=\mu_6$)
            \vspace{5mm}
            \item In the table below, give the weights of the contrast or contrasts you would test to answer the question. There should be one row for each contrast.          
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}  \hline
      ~1~  &   ~2~ & ~3~ &   ~4~ &  ~5~ &   ~6~ \\ \hline
           &       &     &       &      &        \\ 
           &       &     &       &      &        \\ 
           &       &     &       &      &        \\ 
           &       &     &       &      &        \\ \hline
\end{tabular}
\end{center}
%\begin{center}
%\begin{tabular}{|c|c|c|c|c|c|}  \hline
%      ~1~  &   ~2~ & ~3~ &   ~4~ &  ~5~ &   ~6~ \\ \hline
%       1   &   -1  &  0  &    0  &   0  &    0   \\  \hline
%       0   &    1  & -1  &    0  &   0  &    0   \\  \hline
%       0   &    0  &  0  &    1  &  -1  &    0   \\  \hline
%       0   &    0  &  0  &    0  &   1  &   -1   \\  \hline
%\end{tabular}
%\end{center}
        \end{enumerate}
    \end{enumerate}

\item Two contrasts $\mathbf{a}_1^\prime\boldsymbol{\mu}$ and  $\mathbf{a}_2^\prime\boldsymbol{\mu}$ are said to be \emph{orthoganal} if $\mathbf{a}_1^\prime\mathbf{a}_2 = \mathbf{0}$. Show that if the contrasts $c_1$ and $c_2$ are orthoganal and sample sizes are equal, then the estimated contrasts $\widehat{c}_1$ and $\widehat{c}_2$ have zero covariance. Why does this imply that the estimated contrasts are independent if the data are normally distributed?

%%%%%%%%%% Multiple comparisons  

\item Suppose you have data from an experiment with four treatments and a control. If you reject the null hypothesis that all five treatment means are equal, you would like to test all pairwise comparisons at joint significance level 0.05. But comparisons of treatments with the control are more important to detect than comparisons of the treatments with each other. So you decide to carry out the pairwise tests and compute one-at-a-time $p$-values as usual, and then reject $H_0$ with your follow-up tests this way. If $p<0.01$ for a comparison of treatment to control, reject. If $p<0.01/6$ for a comparison of treatment to treatment, reject. Will this procedure protect the family of tests against Type I error at the \emph{joint} 0.05 significance level? Answer Yes or No and show your work. See Bonferroni's inequality on the formula sheet.

\item For the general multiple regression model with fixed independent variables and normal error terms, let $F_1$ denote the test statistic of an initial $F$-test whose null hypothesis imposes $q$ constraints on $\boldsymbol{\beta}$, and let $F_2$ denote the test statistic of a follow-up test whose null hypothesis imposes $s<q$ constraints on $\boldsymbol{\beta}$. We have 
\begin{displaymath}
    F_1 = \frac{SSR-SSR_1}{q \, MSE}
    ~~~~~~~~~
    F_2 = \frac{SSR-SSR_2}{s \, MSE}.
\end{displaymath} 
We reject $H_0$ with a Scheff\'e test if $F_2 > \frac{q}{s}f_\alpha(q,n-p)$, where $f_\alpha(q,n-p)$ is the critical value of the initial test. Show that if the null hypothesis of the initial test is not rejected, then the Scheff\'e test's null hypothesis cannot be rejected either. 

\item A food company wanted to test four different package designs for a new breakfast cereal. The experimental units were twenty stores with approximately equal sales volume. Each package was used in five randomly chosen stores, but a fire in one of the stores caused it to be dropped from the study. The response variable was total sales of the new cereal, in cases. Here is my SAS code.

\begin{verbatim}
/********************** kentonreg.sas **************************/
options linesize=79 pagesize=100 noovp formdlim=' ' nodate;
title 'Kenton Oneway Example From Kutner et al.';

proc format;
     value pakfmt 1 = '3Colour Cartoon'   2 = '3Col No Cartoon'
                  3 = '5Colour Cartoon'   4 = '5Col No Cartoon';

data food;
     infile 'kenton.data';
     input package sales;
     label package = 'Package Design'
           sales   = 'Number of Cases Sold';
     format package pakfmt.;
     if package=1 then p1=1; else p1=0;
     if package=2 then p2=1; else p2=0;
     if package=3 then p3=1; else p3=0;
     if package=4 then p4=1; else p4=0;

proc means n mean stddev;
     class package;
     var sales;





proc reg;
     title2 'Cell means coding';
     model sales = p1 p2 p3 p4 / noint;
     One_vs_2:   test p1=p2;
     One_vs_3:   test p1=p3;
     One_vs_4:   test p1=p4;
     Two_vs_3:   test p2=p3;
     Two_vs_4:   test p2=p4;
     Three_vs_4: test p3=p4;
     Mystery1: test p1+p2=p3+p4;
     Mystery2: test p1+p3=p2+p4;
     Mystery3: test p1-p3=p2-p4;
     Mystery4: test p1=p2, p3=p4;
     Mystery5: test p1=p2=p3=p4;

proc iml;
     title2 'Critical value of initial test';
     numdf = 3;   /* p-1 = Numerator degrees of freedom for initial test */
     dendf = 15;  /* n-p = Denominator degrees of freedom for initial test */
     alpha = 0.05;
     critval = finv(1-alpha,numdf,dendf);  print critval;

\end{verbatim}
The output (somewhat edited) appears below.
\begin{verbatim}

                   Kenton Oneway Example From Kutner et al.                   1

                              The MEANS Procedure

                Analysis Variable : sales Number of Cases Sold
 
                              N
         Package Design     Obs     N            Mean         Std Dev
         ------------------------------------------------------------
         3Colour Cartoon      5     5      14.6000000       2.3021729

         3Col No Cartoon      5     5      13.4000000       3.6469165

         5Colour Cartoon      4     4      19.5000000       2.6457513

         5Col No Cartoon      5     5      27.2000000       3.9623226
         ------------------------------------------------------------
 







                                                                               
 
                   Kenton Oneway Example From Kutner et al.                   2
                               Cell means coding

                               The REG Procedure
                                 Model: MODEL1
                Dependent Variable: sales Number of Cases Sold

                    Number of Observations Read          19
                    Number of Observations Used          19


              NOTE: No intercept in model. R-Square is redefined.

                             Analysis of Variance
 
                                    Sum of           Mean
Source                   DF        Squares         Square    F Value    Pr > F

Model                     4     7183.80000     1795.95000     170.29    <.0001
Error                    15      158.20000       10.54667                     
Uncorrected Total        19     7342.00000                                    


             Root MSE              3.24756    R-Square     0.9785
             Dependent Mean       18.63158    Adj R-Sq     0.9727
             Coeff Var            17.43042                       


                              Parameter Estimates
 
                                            Parameter      Standard
  Variable    Label                  DF      Estimate         Error   t Value

  p1                                  1      14.60000       1.45235     10.05
  p2                                  1      13.40000       1.45235      9.23
  p3                                  1      19.50000       1.62378     12.01
  p4                                  1      27.20000       1.45235     18.73

                             Parameter Estimates
 
               Variable    Label                  DF   Pr > |t|

               p1                                  1     <.0001
               p2                                  1     <.0001
               p3                                  1     <.0001
               p4                                  1     <.0001
 

                                                                               
 
                   Kenton Oneway Example From Kutner et al.                   3
                               Cell means coding

                               The REG Procedure

              Test One_vs_2 Results for Dependent Variable sales
 
                                           Mean
           Source             DF         Square    F Value    Pr > F

           Numerator           1        3.60000       0.34    0.5677
           Denominator        15       10.54667                     
 

              Test One_vs_3 Results for Dependent Variable sales
 
                                           Mean
           Source             DF         Square    F Value    Pr > F

           Numerator           1       53.35556       5.06    0.0399
           Denominator        15       10.54667                     
 
                                                                               
              Test One_vs_4 Results for Dependent Variable sales
 
                                           Mean
           Source             DF         Square    F Value    Pr > F

           Numerator           1      396.90000      37.63    <.0001
           Denominator        15       10.54667                     
 

              Test Two_vs_3 Results for Dependent Variable sales
 
                                           Mean
           Source             DF         Square    F Value    Pr > F

           Numerator           1       82.68889       7.84    0.0135
           Denominator        15       10.54667                     
 
                                                                               
 
              Test Two_vs_4 Results for Dependent Variable sales
 
                                           Mean
           Source             DF         Square    F Value    Pr > F

           Numerator           1      476.10000      45.14    <.0001
           Denominator        15       10.54667                     
 
                                                                               
 
             Test Three_vs_4 Results for Dependent Variable sales
 
                                           Mean
           Source             DF         Square    F Value    Pr > F

           Numerator           1      131.75556      12.49    0.0030
           Denominator        15       10.54667                     
 

              Test Mystery1 Results for Dependent Variable sales
 
                                           Mean
           Source             DF         Square    F Value    Pr > F

           Numerator           1      411.40000      39.01    <.0001
           Denominator        15       10.54667                     
 
                                                                               
 
              Test Mystery2 Results for Dependent Variable sales
 
                                           Mean
           Source             DF         Square    F Value    Pr > F

           Numerator           1       49.70588       4.71    0.0464
           Denominator        15       10.54667                     
 
                                                                               

              Test Mystery3 Results for Dependent Variable sales
 
                                           Mean
           Source             DF         Square    F Value    Pr > F

           Numerator           1       93.18824       8.84    0.0095
           Denominator        15       10.54667                     
 

              Test Mystery4 Results for Dependent Variable sales
 
                                           Mean
           Source             DF         Square    F Value    Pr > F

           Numerator           2       67.67778       6.42    0.0097
           Denominator        15       10.54667                     
 
 



              Test Mystery5 Results for Dependent Variable sales
 
                                           Mean
           Source             DF         Square    F Value    Pr > F

           Numerator           3      196.07368      18.59    <.0001
           Denominator        15       10.54667                     
 
                                                                               
 
                   Kenton Oneway Example From Kutner et al.                  14
                        Critical value of initial test

                                    critval

                                   3.2873821

\end{verbatim}
Please answer these questions based on the output.
    \begin{enumerate}
        \item Write the regression equation.
        \item In terms of your $\beta$ values, what is the null hypothesis of the natural initial test?
        \item Do you reject the null hypothesis of the initial test at $\alpha=0.05$? Give the value of the test statistic and the $p$-value. 
        \item You decide to follow up with all pairwise comparisons, Bonferroni-corrected. Make a $4 \times 4$ table, and put Bonferroni-corrected $p$-values in the upper triangle. Does this allow you to make a claim that one of the package designs is most effective? If Yes, which one is it? 
        \item Now make a similar table for Scheff\'e tests. Instead of putting Scheff\'e-corrected $p$-values, just write the word Yes or No. Naturally you will need a calculator. Are you still able to conclude that one of the designs was most effective? 
        \item Now answer each of the following questions in plain, non-statistical language -- but base your answers on Scheff\'e follow-ups to the initial test. Where possible, draw directional conclusions rather than just answering Yes or No. 
        \begin{enumerate}
            \item Is average response to the packages with cartoons different from average response to the packages without cartoons?
            \item Is average response to the 3-colour packages different from average response to the 5-colour packages?
            \item Does the effect of colour depend on whether or not the package has a cartoon?
            \item Is there an effect of Cartoon for \emph{either} 3-colour packages or 5-colour packages? This is one test. 
        \end{enumerate}
    \end{enumerate}

%%%%%%%%%% 

%%%%%%%%%% 

\end{enumerate}

\vspace{5mm}

\noindent
\begin{center}\begin{tabular}{l}
\hspace{6in} \\ \hline
\end{tabular}\end{center}
This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\href{http://www.utstat.toronto.edu/~brunner/oldclass/305s14} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/305s14}}

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