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I am given the full model, $M_{\tt f}$, with the regression line $$ {\tt response} = \beta_0 + \beta_1{\tt A} + \beta_2{\tt B} + \beta_2{\tt C} + \beta_4{\tt D} + \beta_5{\tt E} + \beta_6{\tt F} + \beta_7{\tt G} + \beta_8 {\tt H} + \varepsilon, \\ \text{where } \varepsilon \sim N(0, \sigma^2) $$ and the following portion of an ${\tt R}$ summary:

                    Estimate Std. Error t value Pr(>|t|)
       (Intercept)  0.669399   1.296381   0.516  0.60690
       A            0.587023   0.087920   6.677 2.11e-09
       B            0.454461   0.170012   2.673  0.00896
       C           -0.019637   0.011173  -1.758  0.08229
       D            0.107054   0.058449   1.832  0.07040
       E            0.766156   0.244309   3.136  0.00233
       F           -0.105474   0.091013  -1.159  0.24964
       G            0.045136   0.157464   0.287  0.77506
       H            0.004525   0.004421   1.024  0.30885

       Residual standard error: 0.7084 on 88 degrees of freedom
       Multiple R-squared:  ????,    Adjusted R-squared:  ????
       F-statistic: 20.86 on 8 and 88 DF,  p-value: ????

I need to calculate the following values, but I am only given the above information.

  1. $R^2$
  2. $R^2_{adj}$
  3. the p-value for the F-statistic

I know that $R^2 = 1 - \text{SSE}/\text{SST}$, and I can calculate $\text{SSE} = \widehat{\sigma}^2 * \text{df}(\text{SSE}) = 0.7084^2 * 88 = 44.16109$, but I can't figure out how to calculate $\text{SST}$. Once I have $\text{SST}$, I can calculate $\text{SSR} = \text{SST} - \text{SSE}$, but I don't know how to calculate the p-value of the F-statistic.

Note:

  • $\text{SSE}$: Sum of Squared Errors
  • $\text{SSR}$: Sum of Squared Residuals or Residual Sum of Squares ($\text{RSS}$)

Thanks in advance for your help!

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  • 3
    $\begingroup$ Work on F-statistic: 20.86 on 8 and 88 DF to get SST. $\endgroup$ – user158565 Dec 10 '18 at 18:31
  • 1
    $\begingroup$ $F = \frac{SST - SSE}{p} \Big/ \frac{SSE}{n-p-1} = 127.9066$. From that, I can find $R^2 = 1 - \frac{SSE}{SST} = 0.655$. $\endgroup$ – inkalchemist1994 Dec 10 '18 at 18:48
  • $\begingroup$ For p value, search for F distribution calculator on internet. $\endgroup$ – user158565 Dec 10 '18 at 20:17
  • $\begingroup$ I don't want to just get the answer; I want to understand where the answer comes from. $\endgroup$ – inkalchemist1994 Dec 10 '18 at 21:51
  • $\begingroup$ Then begin from $\epsilon \sim N(0,\sigma^2)$, 1) derive the distribution of SSR (hint: close to chi-square). 2) derive the distribution of SSE (hint: close to chi-square). 3) prove that SSR and SSE are independent. 4) Derive the distribution of ratio of SSR/df1 over SSE/df2. (hint: F distribution). 5) find the CDF or PDF of F distribution. 6) calculate $\int_{127.9066}^{\infty}f(x)dx$, where $f(x)$ is pdf of F distribution $\endgroup$ – user158565 Dec 11 '18 at 0:13

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