Statistics 333 Chernobyl Example Spring 2003This is an annotated analysis of a data set on the concentrations of radioactive cesium in mushrooms and soil sampledfrom 17 wooded locations in central Italy at times within a couple years of the nuclear reactor accident in Chernobyl. Thepurpose of the study was to examine the uptake of radioactive cesium by mushrooms from the surrounding soil.Here is the actual data set with concentrations of the radioactive isotopes of cesium (in Bq/kg) in soil and mushrooms.> ex0818MUSHROOM SOIL1 1 332 9 553 14 1384 17 3195 20 4156 17 4257 14 4428 15 4759 34 27910 41 32911 46 8212 49 8613 53 5514 60 6015 79 14416 99 29217 190 1310Here is a plot of the data with a regression line fitted and a residual plot versus the fitted values.> par(mfrow = c(1, 2))> plot(SOIL, MUSHROOM, pch = 16)> fit1 <- lm(MUSHROOM ~ SOIL)> abline(fit1)> plot(fitted(fit1), residuals(fit1), pch = 16)> abline(h = 0)●●●●●●●●●●●●●●●●●0 400 800 12000 50 100SOILMUSHROOM●●●●●●●●●●●●●●●●●20 60 100 140−40 0 20fitted(fit1)residuals(fit1)Bret Larget March 12, 2003Statistics 333 Chernobyl Example Spring 2003We notice that the 17th observation is an outlier. It has unusually large measurements in both variables. The residualis not that unusual, but this can be an artifact. This one point will be highly influential and will tend to pull the regressionline close to it to avoid having a single enormous squared residual.Here is a summary of the regression and a 95% confidence interval for the slope.> sum1 <- summary(fit1)> sum1Call:lm(formula = MUSHROOM ~ SOIL)Residuals:Min 1Q Median 3Q Max-47.279 -30.319 -9.483 31.000 54.271Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) 16.72569 12.41954 1.347 0.19807SOIL 0.09590 0.02993 3.205 0.00591 **---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 36.56 on 15 degrees of freedomMultiple R-Squared: 0.4064, Adjusted R-squared: 0.3668F-statistic: 10.27 on 1 and 15 DF, p-value: 0.005909> df1 <- df.residual(fit1)> t.crit <- qt(0.975, df1)> t.crit[1] 2.131450> t.crit * (coef(sum1))[2, 2][1] 0.06378854A 95% confidence interval for the slope β1is 0.096 ± 0.064.We ought to be concerned about the effect of the outlier. One possibility to consider is to see if the estimated regressionline is substantially different if the point is excluded from the analysis.> par(mfrow = c(1, 2))> keep <- 1:16> plot(SOIL[keep], MUSHROOM[keep], pch = 16)> fit2 <- lm(MUSHROOM[keep] ~ SOIL[keep])> abline(fit2)> abline(fit1, lty = 2)> plot(fitted(fit2), residuals(fit2), pch = 16)> abline(h = 0)Bret Larget March 12, 2003Statistics 333 Chernobyl Example Spring 2003●●●●●●●●●●●●●●●●100 3000 20 60 100SOIL[keep]MUSHROOM[keep]●●●●●●●●●●●●●●●●30 35 40−40 0 20 60fitted(fit2)residuals(fit2)Notice that the new regression line is substantially different from the previous regression line.Here is a summary of the regression and a new 95% confidence interval for the slope.> sum2 <- summary(fit2)> sum2Call:lm(formula = MUSHROOM[keep] ~ SOIL[keep])Residuals:Min 1Q Median 3Q Max-41.658 -13.938 -4.061 9.744 65.908Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) 43.87726 12.25571 3.580 0.00301 **SOIL[keep] -0.03693 0.04454 -0.829 0.42085---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 27.76 on 14 degrees of freedomMultiple R-Squared: 0.04682, Adjusted R-squared: -0.02126F-statistic: 0.6877 on 1 and 14 DF, p-value: 0.4208> df2 <- df.residual(fit2)> t.crit <- qt(0.975, df2)> t.crit[1] 2.144787> t.crit * (coef(sum2))[2, 2][1] 0.0955225Bret Larget March 12, 2003Statistics 333 Chernobyl Example Spring 2003A 95% confidence interval for the slope β1is −0.037 ± 0.096.The inferences on the relationship between radioactive cesium concentration in soil and in mushrooms is now completelydifferent. With the outlier included, there is strong evidence that the concentration of radioactive cesium in mushroomsincreases as the concentration in the soil increases. When this point is excluded, there is no evidence of a relationship.All we can really conclude is that for the one data point in which the soil concentration was quite high, the mushroomconcentration was very high as well. For lower concentrations, there is very little evidence of a strong relationship.An alternative analysis would be to analyze concentrations on the log scale. Here are the plots.> par(mfrow = c(1, 2))> plot(log(SOIL), log(MUSHROOM), pch = 16)> fit3 <- lm(log(MUSHROOM) ~ log(SOIL))> abline(fit3)> plot(fitted(fit3), residuals(fit3), pch = 16)> abline(h = 0)●●●●●●●●●●●●●●●●●4 5 6 70 1 2 3 4 5log(SOIL)log(MUSHROOM)●●●●●●●●●●●●●●●●●2.5 3.0 3.5 4.0−2 −1 0 1fitted(fit3)residuals(fit3)On this scale, the sample with the smallest soil concentration is a bit of an outlier, but we do not see much of a relationship.The slope is not significant on the log scale.> summary(fit3)Call:lm(formula = log(MUSHROOM) ~ log(SOIL))Residuals:Min 1Q Median 3Q Max-2.55598 -0.70057 0.05032 1.09945 1.28070Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) 1.0490 1.4896 0.704 0.492log(SOIL) 0.4310 0.2804 1.537 0.145Residual standard error: 1.136 on 15 degrees of freedomMultiple R-Squared: 0.1361, Adjusted R-squared: 0.07847F-statistic: 2.362 on 1 and 15 DF, p-value: 0.1451Bret Larget March 12,