Errata (excluding minor typos) for the book.

If you notice an error that is not mentioned in the errata below, submit an issue at https://github.com/avehtari/ROS-Examples/issues or send an email.

- p. 39. the weight and metabolisim values in the text did not match with values in Figure 3.4. The values in the text have been made to match more accurately the values in the Figure 3.4. See more details in Schmidt-Nielsen (1984).
- p. 106, Figure 8.1a “\(p(a, b | y)\)” \(\rightarrow\) “\(p(y | a, b)\)” (thanks to Jun Tsunematsu)
- p. 146, \(\mbox{RSS} = \sum_{i=1}^n(y_i - X\hat{\beta})^2\) should be \(\mbox{RSS} = \sum_{i=1}^n(y_i - X_i\hat{\beta})^2\) (thanks to Ravi Shroff)
- p. 147, \(W=\mbox{Diag}(w)\) should be \(W=\mbox{Diag}(w^{-1})\) (thanks to Ravi Shroff)
- p. 150, ex 10.2b “for the treatment group and the control group,” \(\rightarrow\) “for the treatment group,” (thanks to Jakob Staudt)
- p. 167, the code line
`y_rep[s,1] <- y[1]`

should be`y_rep[s,1] <- unemp$y[1]`

(thanks to Ravi Shroff) - p. 181, ex 11.5(b) “Figure 10.2.” should be “Figure 11.2.” (thanks to Ravi Shroff)
- p. 187, “a procedure that is equivalent to centering and rescaling is to leave” \(\rightarrow\) “we can get the same inferences for the coefficients other than the intercept by leaving” (thanks to Ravi Shroff)
- p. 209, “\(\frac{p_0}{D-p_0}\frac{\sigma}{\sqrt{n}}\), where \(p_0\) is the expected number of relevant predictors” \(\rightarrow\) “\(\frac{p_0}{p-p_0}\frac{\sigma}{\sqrt{n}}\), where \(p\) is the number of predictors, \(p_0\) is the expected number of relevant predictors” (thanks to Roberto Viviani)
- p. 231, \(\log(-.595)\) \(\rightarrow\) \(\log(0.595)\) (thanks to Daniel Timar)
- p. 267
`fit_nb[[k]] <- stan_glm(y ~ x, family=neg_binomial_2(link="log"), data=fake, refresh=0)`

\(\rightarrow\)`fit_nb[[k]] <- stan_glm(y ~ x, family=neg_binomial_2(link="log"), data=fake_nb[[k]], refresh=0)`

(thanks to Brian Bucher) - p. 268,
`offset=log(exposure)`

\(\rightarrow\)`offset=log(exposure2)`

(thanks to A. Solomon Kurz) - p. 269,
`y_rep_1 <- posterior_predict(fit_1)`

\(\rightarrow\)`yrep_1 <- posterior_predict(fit_1)`

(thanks to Darci Kovacs) - p. 285, The numbers for the earnings model on p.285 should be the same as on p. 284 (thanks to David Galley)
- p. 295, \(n=(2.8* 0.49/0.1)^2=196\) \(\rightarrow\) \(n=(2.8* 0.5/0.1)^2=196\), where \(0.5\) is used as a conservative estimate of the standard deviation (thanks to Solomon A. Kurz)
- p. 299, \(0.5/1.15 = 0.43\) \(\rightarrow\) \(0.5/1.25 = 0.4\) (thanks to Solomon A. Kurz)
- p. 300, The paragraph starting “We illustrate with the example of the survey earnings and height discussed in Chapter 4.” and the next two paragraphs have been edited to:

We illustrate with the survey of earnings and height discussed in Chapter 12. The coefficient for the sex-earnings interaction in model (12.2) is plausible (a positive interaction, implying that an extra inch of height is worth 2% more for men than for women), with a standard error of 1%.

Extracting another significant figure from the fitted regression yields an estimated interaction of 0.018 with standard error 0.015. How large a sample size would have been needed for the coefficient on the interaction to be “statistically significant” at the conventional 95% level? A simple calculation uses the fact that standard errors are proportional to \(1/\sqrt{n}\). For a point estimate of 0.018 to be two standard errors from zero, it would need a standard error of 0.009, which would require the sample size to be increased by a factor of \((0.015/0.009)^2\). The original survey had a sample of 1629; this implies a required sample size of \(1629*(0.015/0.009)^2=4500\).

To perform a power calculation for this hypothetical larger survey of 4500 people, we could suppose that the true \(\beta\) for the interaction is equal to 0.018 and that the standard error is as we have just calculated. With a standard error of 0.009 the estimate from the regression would then be conventionally “statistically significant” only if \(\hat{\beta}>0.018\) (or, in other direction, if \(\hat{\beta}< -0.018\), but that latter possibility is highly unlikely given our assumptions). If the true coefficient \(\beta\) is 0.018, then we would expect \(\hat{\beta}\) to exceed 0.018, and thus achieve statistical significance, with a probability of \(\frac{1}{2}\)—that is, 50% power. To get 80% power, we need the true \(\beta\) to be 2.8 standard errors from zero, so that there is an 80% probability that \(\hat{\beta}\) is at least 2 standard errors from zero. If \(\beta=0.018\%\), then its standard error would have to be no greater than \(0.018/2.8\), so that the survey would need a sample size of \(1629*(2.8*0.015/0.018)^2=9000\). (thanks to comment by Solomon A. Kurz, and typo fix by Patrick Wen) - p. 304 “only 44% of their children were girls” \(\rightarrow\) “only 48% of their children were girls” (thanks to comment by Solomon A. Kurz)
- p. 375, three regression equations (unnumbered, 19.2 and 19.3) are missing $+ $ from the beginning of RHS (right after \(=\)). (thanks to Junhui Yang)
- p. 407, \(\sum_{k=1}^K(X_{ik}-X_{jk})^2\) \(\rightarrow\) \((\sum_{k=1}^K(X_{ik}-X_{jk})^2)^{1/2}\) (thanks to Stefan Gehrig)
- p. 324, “90% of African Americans” \(\rightarrow\) “88% of African Americans” (thanks to Solomon A. Kurz)

- p. 11, the summary of the treatment-control comparison says “the treated units were 4.8 points higher than the controls, \(\bar{y} = 31.7\) under the treatment and \(\bar{y} = 25.5\) for the controls.” The difference in shown means is 6.2 and not 4.8. Not that these values change when rerunning the simulation. (thanks Desislava Petkova)
- p. 15, “multiplicative factor; see Exercise 3.8.” \(\rightarrow\) “multiplicative factor; see Exercise 4.8.” (thanks to Omri Har-Shemesh and Juliette Unwin)
- p. 39, in the second sentence of the first full paragraph, “Figure 3.4 displays data on log metabolic rate vs. body mass indicating…” \(\rightarrow\) “log metabolic rate vs. log body mass” (thanks Ravi Shroff)
- p. 39, “The slope of the line is then 7.4/10 = 7.4.” \(\rightarrow\) “The slope of the line is then 7.4/10 = 0.74.” (thanks to Jacob Warren)
- p. 42, the last line, in “Linear transformations” section “Exercise 3.5” \(\rightarrow\) “Exercise 3.6”. (thanks Ed Berry)
- p. 43, “\(\sigma_a,\sigma_b\)” \(\rightarrow\) \(\sigma_u,\sigma_v\) (thanks to Braden Scherting)
- p. 46, “exactly 10 000 votes” \(\rightarrow\) “exactly 100 000 votes” (thanks to Gal Matijevic)
- p. 54, in “Comparisons, visual and numerical” subsection, “Figure 4.2” in the first sentence should be “Figure 4.3.” (thanks Ravi Shroff)
- p. 55,
`se_weighted_avg <- sqrt(sum(W*se)^2)`

\(\rightarrow\)`se_weighted_avg <- sqrt(sum((W*se)^2))`

(thanks to Desislava Petkova) - p. 57, last line:
`n_C+n_T, 2`

\(\rightarrow\)`n_C+n_T-2`

(thanks Justin Reppert) - p. 61, “\(T(y;\phi(y))\), for \(j = 1,\dots, J\)” \(\rightarrow\) “\(T(y;\phi_j(y))\), for \(j = 1,\dots, J\)” (thanks to Kenneth Tay)
- p. 73,
`1.483*median(abs(y - median(z)))`

\(\rightarrow\) 1.483*median(abs(z - median(z))) - p. 73,
`quantile(z, 0.25, 0.75)`

\(\rightarrow\) quantile(z, c(0.25, 0.75)) and`quantile(z, 0.025, 0.975)`

\(\rightarrow\) quantile(z, c(0.025, 0.975)) (thanks to Justin Reppert) - p. 83, Fig 6.1 “y = 0.49 + 0.27 * x” \(\rightarrow\) “y = 0.4 + 0.27 * x”
- p. 83, Fig 6.2 Estimate for \(\sigma\) should be 0.49 (not 0.43)
- p. 84, “which returns the value R^2 = 0.10, meaning that the linear model accounts for only 9% of the variance”. The “9%” should be “10%”.
- p. 105, just before Maximum likelihood section “see Exercise 7.1” \(\rightarrow\) “see Exercise 8.1” (thanks to Jacob Warren)
- p. 116, the correct code should have
`y_linpred <- a + b*as.numeric(new)`

and`y_pred <- a + b*as.numeric(new) + rnorm(n_sims, 0, sigma)`

(thanks to Nathan Hoffmann) - p. 128, “Do Exercise 3.8” \(\rightarrow\) “Do Exercise 4.8” (thanks to Omri Har-Shemesh and Juliette Unwin)
- p. 131, “the average or expected difference in outcome \(y_k\)” \(\rightarrow\) “… \(y\)” (thanks to Yoav Kessler)
- p. 143, the correct values for mean and sd in the last column of Figure 10.7 are 260.1 and 2.5 (thanks to Eam O’Brien)
- p. 143, 3rd line from bottom the correct values for mean and sd mentioned in the text are 260.1 and 2.5 (thanks to Eam O’Brien)
- p. 159, “as in Figure 11.2” \(\rightarrow\) “as in Figure 11.3” (thanks to Eam O’Brien)
- p. 170, “Even though \(R^2\) is much lower in Figure 11.16a, the model fits the data just as well as in Figure 11.16b.” \(\rightarrow\) “Even though \(R^2\) is much lower in Figure 11.16b, the model fits the data just as well as in Figure 11.16a.”
- p. 174, “and so the log score is \(-\frac{1}{2}\log\sigma\)” \(\rightarrow\) “and so the log score is \(-\log\sigma\)” (thanks to Matěj Grabovsky)
- p. 179,
`fit_2 <- update(fit_2, prior=hs())`

\(\rightarrow\)`fit_2 <- update(fit_1, prior=hs())`

(thanks to Omri Har-Shemesh) - p. 183, \(\$1600* 3.8=\$63* 97=\$101\,000\,000* 0.000\,060=\$610\) \(\rightarrow\) \(\$1600* 3.8=\$63* 97=\$101\,000\,000* 0.000\,060=\$6100\) (thanks to Ravi Shroff)
- p. 190 “an expected positive difference of about 5% in the outcome variable” \(\rightarrow\) “an expected positive difference of about 6% in the outcome variable”
- p. 208,
`/sqrt(0.3*26)`

\(\rightarrow\)`*sqrt(0.3/26)`

(this is correct in the code in the web page, typo fix in the book thanks to Eugenia Migliavacca) p. 208, Figure 12.11 all top row subplots show the prior distribution for the regularized horseshoe prior. The correct subplot are produced by the R code in Student example and shown below (thanks to Zhengchen Cai for reporting the issue)

- p. 213, Ex 12.11, “\(\beta\) is 0.3” \(\rightarrow\) \(\beta\) is \(-0.3\) (thanks t o Justin Gross)
- p. 217, First line in section 13.1 “The logistic function, logit(x)” \(\rightarrow\) “The logit function, logit(x)” (thanks to VicentModesto)
- p. 219, “The inverse logistic function is curved” \(\rightarrow\) “The inverse logit function is curved” (thanks to VicentModesto)
- p. 241-242, in paragraphs starting “The steps go…” and “Figure 14.2…”: \((4.0/4.1)x_1\) \(\rightarrow\) \((4.4/4.1)x_1\) (thanks to Doug Davidson)
- p. 288, Chapter 15 Bibliographic note is missing the reference to the RiskyBehavior data: “The HIV risk data (RiskyBehavior) used in exercises comes fromcomes from El-Bassel et al (2003).” Full reference: El-Bassel, N., Witte, S. S., Gilbert, L., Wu, E., Chang, M., Hill, J., and Steinglass, P. (2003). The efficacy of a relationship-based HIV/STD prevention program for heterosexual couples.
*American Journal of Public Health*,**93**, 963–969. - p. 314, Following Equation (17.1) “… did not include enough Republicans.” \(\rightarrow\) “… included too many Democrats.” (thanks to Mark Fisher)
- p. 316, Figure 17.2 the values in the column \(N_j/N\) should be \(0.16, 0.17, 0.19, 0.17, 0.16, 0.15\) as in the code on the same page. (thanks to Noah Silbert)
- p. 344, “…Figure 18.1 yields a naive estimated treatment effect of -12.5.” \(\rightarrow\) “…Figure 18.1 yields a naive estimated treatment effect of +12.5.” (thanks to Michael McLaren)
- p. 345, “difference in means, \(155-147.5\)” \(\rightarrow\) “difference in means, \(147.5 - 155\)” (thanks to Michael McLaren)
- p. 345, at the bottom “More to the point, the \(y_i^0\)’s are 140, 140, 150, and 170 in the
*control*group and 150, 160, 160, and 170 in the*treatment*group.” \(\rightarrow\) “More to the point, the \(y_i^0\)’s are 140, 140, 150, and 170 in the*treatment*group and 150, 160, 160, and 170 in the*control*group.” (thanks to Ravi Shroff) - p. 349 “data presented in Figure 18.4” should be “data presented in Figure 18.3”
- p. 371, “in Figure 19.4a” \(\rightarrow\) “in Figure 19.4b” (thanks to Kenneth Tay)
- p. 385, equation number (20.1) should be for the second equation on the page \(y_i = \beta^*_0 + \beta^*_1z_i + \epsilon^*_i\) (thanks to Matěj Grabovsky)
- p. 397, last line, in the standard error equation 1738 \(\rightarrow\) 1820, 789 \(\rightarrow\) 837, and 1.01 \(\rightarrow\) 1.04 (thanks to Matěj Grabovsky)
- p. 435, equation 21.7: “\(\beta_2 *\mbox{math pre-test}_i\)” \(\rightarrow\) “\(\beta_3 *\mbox{math pre-test}_i\)” (thanks to Zhengchen Cai)
p. 447, “populations homogeneous in \(w_i\)” \(\rightarrow\) “populations homogeneous in \(x_i\)” (thanks to Kenneth Tay)