{"id":1194,"date":"2015-08-27T10:36:47","date_gmt":"2015-08-27T09:36:47","guid":{"rendered":"http:\/\/pcool.dyndns.org:8080\/statsbook\/?page_id=1194"},"modified":"2025-06-24T23:02:12","modified_gmt":"2025-06-24T22:02:12","slug":"answers-curve-fitting","status":"publish","type":"page","link":"https:\/\/pcool.dyndns.org\/index.php\/answers-curve-fitting\/","title":{"rendered":"Answers Curve Fitting"},"content":{"rendered":"\n

 <\/p>\n\n\n\n

The data is also provided in Q8-1.rda <\/a>(the data frame is called Q8). To call the data in the console:<\/p>\n\n\n\n

Q8<\/em><\/span>\n  x    y<\/em><\/span>\n1 3 1.20<\/em><\/span>\n2 4 1.40<\/em><\/span>\n3 5 1.60<\/em><\/span>\n4 6 1.75<\/em><\/span>\n5 7 1.85<\/em><\/span><\/code><\/pre>\n\n\n\n
    \n
  1. Plot the data, draw the regression line and estimate the equation of the line:<\/li>\n<\/ol>\n\n\n\n
    ggplot() + <\/span><\/em>\ngeom_point(aes(x = x,y = y), data=Q8) + <\/span><\/em>\ngeom_smooth(aes(x = x,y = y),data=Q8,method = 'lm') + <\/span><\/em>\ntheme_bw()<\/span><\/em><\/code><\/pre>\n\n\n\n
    \"\"<\/figure>\n\n\n\n
    Fit the regression line:<\/p>\n\n\n\n
    fit<-lm(y~x,data=Q8)<\/em><\/span>\nfit<\/em><\/span>\nCall:<\/em><\/span>\nlm(formula = y ~ x, data = Q8)<\/em><\/span>\nCoefficients:<\/em><\/span>\n(Intercept)            x  <\/em><\/span>\n      0.735        0.165 <\/em> <\/span><\/code><\/pre>\n\n\n\n
    <\/p>\n\n\n\n
    The equation of the regression line therefore is:<\/p>\n\n\n\n
    y = 0.735 + 0.165 \u00d7 x<\/strong><\/p>\n\n\n\n
    2. What is the correlation coefficient?<\/p>\n\n\n\n
    cor(Q8$x,Q8$y,method='pearson')<\/em><\/span>\n[1] 0.9913889<\/em><\/span><\/code><\/pre>\n\n\n\n
    The correlation coefficient therefore is 99%.<\/p>\n\n\n\n
    3. Interpolate the y-value for x = 5.5<\/p>\n\n\n\n
    y(x = 5.5) = 0.735 + 0.165 \u00d7 5.5<\/strong> = 1.6425<\/strong><\/p>\n\n\n\n
    4. Extrapolate the y-values for x = 0.1 and x = 15<\/p>\n\n\n\n
    y(x = 0.1) = 0.735 + 0.165 \u00d7 0.1<\/strong> = 0.7515<\/strong><\/p>\n\n\n\n
    y(x = 15) = 0.735 + 0.165 \u00d7 15<\/strong> = 3.21<\/strong><\/p>\n\n\n\n
    The data is also provided in Q8-2.rda <\/a>(the data frame is called Q8Extended). To call the data in the console:<\/p>\n\n\n\n
    Q8Extended<\/em><\/span>\n      x     y<\/em><\/span>\n1   0.1 -1.25<\/em><\/span>\n2   0.2 -0.70<\/em><\/span>\n3   1.0  0.40<\/em><\/span>\n4   2.0  0.90<\/em><\/span>\n5   3.0  1.20<\/em><\/span>\n6   4.0  1.40<\/em><\/span>\n7   5.0  1.60<\/em><\/span>\n8   6.0  1.75<\/em><\/span>\n9   7.0  1.85<\/em><\/span>\n10  8.0  1.95<\/em><\/span>\n11  9.0  2.05<\/em><\/span>\n12 10.0  2.10<\/em><\/span>\n13 12.0  2.25<\/em><\/span>\n14 14.0  2.35<\/em><\/span>\n15 15.0  2.40 <\/em><\/span><\/code><\/pre>\n\n\n\n
    5. Plot these data in a graph.<\/p>\n\n\n\n
    To create a scatterplot<\/a> to evaluate the relation between the two variables (without a regression line):<\/p>\n\n\n\n
    ggplot() + <\/em><\/span>\ngeom_point(aes(x = x,y = y),data=Q8Extended) + <\/em><\/span>\ntheme_bw()<\/em><\/span><\/code><\/pre>\n\n\n\n
    \"\"<\/figure>\n\n\n\n
    6. What is the relation between x and y and what is the value of the correlation coefficient?<\/p>\n\n\n\n
    A linear regression is clearly hopeless:<\/p>\n\n\n\n
    ggplot() + <\/em><\/span>\ngeom_point(aes(x = x,y = y),data=Q8Extended) + <\/em><\/span>\ngeom_smooth(aes(x = x,y = y),data=Q8Extended,method = 'lm') + theme_bw()<\/em><\/span><\/code><\/pre>\n\n\n\n
    \"\"<\/figure>\n\n\n\n
    x and y appear to have a logarithmic relation and the general equation of the regression line is:<\/p>\n\n\n\n
    y = b + a \u00d7 log(x)<\/p>\n\n\n\n
    or:<\/p>\n\n\n\n
    ggplot() + <\/span><\/em>\ngeom_point(aes(x = log<\/strong>(x),y = y),data=Q8Extended) + <\/span><\/em>\ngeom_smooth(aes(x = log<\/strong>(x),y = y),data=Q8Extended,method = 'lm') + theme_bw()<\/span><\/em><\/code><\/pre>\n\n\n\n
    \"\"<\/figure>\n\n\n\n
    The equation of the regression line is:<\/p>\n\n\n\n
    fit<-lm(y~log(x),data=Q8Extended)<\/em><\/span>\nfit<\/em><\/span>\nCall:<\/span><\/em>\nlm(formula = y ~ log(x), data = Q8Extended)<\/span><\/em>\n\nCoefficients:\n(Intercept)       log(x)\n0.4271       0.7276  <\/span> <\/em><\/code><\/pre>\n\n\n\n
    Or:<\/p>\n\n\n\n
    y = 0.4271 + 0.7276 \u00d7 log(x)<\/strong><\/p>\n\n\n\n
    Alternatively, you could plot the data on a logarithmic x axis:<\/p>\n\n\n\n
    ggplot(data=Q8Extended, aes(x=x, y=y)) + <\/em>\ngeom_point() + <\/em>\ncoord_trans(x = \"log10\") + <\/em>\ntheme_bw()<\/em><\/mark><\/code><\/pre>\n\n\n\n
    \"\"<\/figure>\n\n\n\n
    To find the value of the correlation coefficient:<\/p>\n\n\n\n
    cor(log(Q8Extended$x),Q8Extended$y,method='pearson')<\/em><\/span>\n[1] 0.9997933<\/span><\/em><\/code><\/pre>\n\n\n\n
    7. What are the y-values for x = 0.1 and x = 15?<\/p>\n\n\n\n
    As described under 6, the equation of the regression line is:<\/p>\n\n\n\n
    y = 0.4271 + 0.7276 \u00d7 log(x)<\/p>\n\n\n\n
    Therefore,<\/p>\n\n\n\n
    y(x = 0.1) = 0.4271 + 0.7276 \u00d7 log(0.1)<\/strong> \u2248 -1.25<\/strong><\/p>\n\n\n\n
    y(x = 15) = 0.4271 + 0.7276 \u00d7 log(15)<\/strong> \u2248 2.40<\/strong><\/p>\n\n\n\n
    This question illustrates again the danger of extrapolating data!<\/p>\n\n\n\n
    8. The scatterplot and line x=y can be created with the following command in the R console:<\/p>\n\n\n\n
    ggplot() + <\/em><\/span>\ngeom_point(aes(x = Lafayette,y = iPhone),data=goniometer) + <\/em><\/span>\ntheme_bw() + <\/em><\/span>\ngeom_abline(data=goniometer, intercept = 0.0,slope = 1.0,<\/em><\/span>\n  colour = '#339900', linetype = 2)<\/em><\/span><\/code><\/pre>\n\n\n\n
    \"\"<\/figure>\n\n\n\n
    9. To calculate the Pearson correlation coefficient:<\/p>\n\n\n\n
    cor(goniometer$Lafayette,goniometer$iPhone,method=\"pearson\")<\/em><\/span>\n[1] 0.9473263<\/em><\/span><\/code><\/pre>\n\n\n\n
    Therefore, Pearson’s correlation coefficient is 95%.<\/p>\n\n\n\n
    10. To calculate the ICC:<\/p>\n\n\n\n
    library(irr)<\/span><\/em>\nicc(goniometer,model=\"twoway\",type=\"agreement\")<\/span><\/em>\n Single Score Intraclass Correlation<\/span><\/em>\n   Model: twoway <\/span><\/em>\n   Type : agreement <\/span><\/em>\n   Subjects = 60 <\/span><\/em>\n     Raters = 2 <\/span><\/em>\n   ICC(A,1) = 0.948<\/strong><\/span><\/em>\n F-Test, H0: r0 = 0 ; H1: r0 > 0 <\/span><\/em>\n F(59,59.8) = 37 , p = 1.6e-31 <\/span><\/em>\n 95%-Confidence Interval for ICC Population Values:<\/span><\/em>\n  0.914 < ICC < 0.968<\/span><\/em><\/code><\/pre>\n\n\n\n
    Therefore, the ICC is 95%.<\/p>\n\n\n\n
    <\/p>\n","protected":false},"excerpt":{"rendered":"
      The data is also provided in Q8-1.rda (the data frame is called Q8). To call the data in the console: Fit the regression line: The equation of the regression line therefore is: y = 0.735 + 0.165 \u00d7 x 2. What is the correlation coefficient? The correlation coefficient therefore is 99%. 3. Interpolate the […]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1194","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/pages\/1194","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/comments?post=1194"}],"version-history":[{"count":4,"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/pages\/1194\/revisions"}],"predecessor-version":[{"id":4385,"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/pages\/1194\/revisions\/4385"}],"wp:attachment":[{"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/media?parent=1194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}