{"id":2151,"date":"2019-11-15T19:05:51","date_gmt":"2019-11-16T00:05:51","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/simplestats\/?post_type=chapter&#038;p=2151"},"modified":"2019-11-15T19:23:49","modified_gmt":"2019-11-16T00:23:49","slug":"10-2-3-hypothesis-testing-of-b","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/simplestats\/chapter\/10-2-3-hypothesis-testing-of-b\/","title":{"raw":"10.2.3 Hypothesis Testing and Confidence Intervals for the Regression Coefficient","rendered":"10.2.3 Hypothesis Testing and Confidence Intervals for the Regression Coefficient"},"content":{"raw":"[latexpage]\r\n\r\nTo test the regression coefficient\u00a0<em>b<\/em> for significance we have the following hypotheses:\r\n<ul>\r\n \t<li>H<sub>0<\/sub>: The independent variable <em>x<\/em> has no effect on the dependent variable <em>y<\/em> (i.e., the variables are not associated); <em>\u03b2<\/em>=0.<\/li>\r\n \t<li>H<sub>a<\/sub>: The independent variable<em> x<\/em> has an effect on the dependent variable<em> y<\/em> (i.e., the variables are associated);\u00a0<em>\u03b2<\/em>\u22600[footnote]Note that I am using causal language here with the assumption that the conditions for causality are met. Theirs is a separate investigation. In and of itself, finding a significant effect of <em>x<\/em> on <em>y<\/em> does not itself establish that changes in<em> x<\/em> <em>cause<\/em>\u00a0changes in\u00a0<em>y<\/em>.[\/footnote].<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n<strong>After calculating<em> t<sub>b<\/sub><\/em> with <em>df<\/em>=<em>N<\/em>-2 and finding its associated <em>p<\/em>-value, we then compare the <em>p<\/em>-value to the pre-selected significance level\u00a0<em>\u03b1<\/em>. As usual, when<em> p<\/em>\u2264<em>\u03b1<\/em>, we reject the null hypothesis, and have enough evidence to deem the regression coefficient <em>b<\/em> statistically significant. If, on the contrary, <em>p<\/em>&gt;<em>\u03b1<\/em>, we fail to reject the null hypothesis and therefore conclude that at present there is no evidence to suggest an effect of <em>x<\/em> on <em>y<\/em>.<\/strong>\r\n\r\n&nbsp;\r\n\r\nAgain, similarly to other statistics, we can <strong>calculate confidence intervals for <em>b<\/em>, so that we can report the size of the effect with a specific level of certainty<\/strong>. For example, the 95% confidence interval for the regression coefficient <em>b<\/em> is:\r\n<ul>\r\n \t<li>95% CI: $b\\pm 1.96\\times s_b$<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\nTo illustrate, let's revisit our example about the effect of parental education on their offspring education. (Don't worry,\u00a0<span style=\"text-indent: 18.6667px;font-size: 14pt\">with <em>N<\/em>=1, 686\u00a0<\/span><span style=\"text-indent: 1em;font-size: 14pt\">I will not offer you a calculation by hand: SPSS is there for us.)\u00a0\u00a0<\/span>\r\n\r\n&nbsp;\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\"><em>Example 10.4 Effect of Parental Years of Schooling on Respondent's Years of Schooling (GSS 2018)<\/em><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n&nbsp;\r\n\r\nWe already examined the association between parental and offspring education through the correlation coefficient <em>r<\/em> and found it to be moderately weak at 0.413, and statistically significant at\u00a0<em>\u03b1<\/em>=0.01. Can we do better, however, and estimate the effect of each additional year of parental schooling on the schooling of the respondents?\r\n\r\n&nbsp;\r\n\r\nAgain, we use data from the U.S.\u00a0<em>GSS 2018<\/em> (NORC, 2019).\u00a0 Our sample is <em>N<\/em>=1,686, and <strong>our hypotheses are:<\/strong>\r\n<ul>\r\n \t<li>H<sub>0<\/sub>: Father's education has no effect on respondent's education;\u00a0<em>\u03b2<\/em>=0.<\/li>\r\n \t<li>H<sub>a<\/sub>: Father's education has an effect on respondent's education;\u00a0<em>\u03b2<\/em>\u22600.<\/li>\r\n<\/ul>\r\n<strong>The regression model is:<\/strong>\r\n\r\n&nbsp;\r\n\r\n$\\textrm{years of schooling}=y=a+bx+e=a+b(\\textrm{years of parental schooling})+e$\r\n\r\n&nbsp;\r\n\r\n<strong>Our predicted values are:<\/strong>\r\n\r\n&nbsp;\r\n\r\n$\\textrm{predicted years of schooling} =\\hat{y}=a+bx=a+b(\\textrm{years of parental schooling})$\r\n\r\n&nbsp;\r\n\r\nFigure 10.3 plots the association and Table 10.4 show the relevant SPSS output.\r\n\r\n&nbsp;\r\n\r\nFigure 10.3\u00a0<em>Linear Regression of Respondent's Years of Schooling and Father's Years of Schooling<\/em>\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/scatterplot-educ-paeduc-line.png\" alt=\"\" width=\"462\" height=\"370\" class=\"wp-image-1371 size-full aligncenter\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<em>Table 10.4 Linear Regression of Respondent's Years of Schooling and Father's Years of Schooling<\/em>\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/regression-table-educ-paeduc.png\" alt=\"\" width=\"812\" height=\"178\" class=\"wp-image-1370 size-full aligncenter\" \/>\r\n\r\n&nbsp;\r\n\r\nThat is, SPSS has calculated the constant (or <em>Y<\/em>-intercept) <em>a<\/em> and the regression coefficient <em>b<\/em> in such a way as to minimize the residuals:\r\n<ul>\r\n \t<li><em>a<\/em> = 10.67<\/li>\r\n \t<li><em>b<\/em> = 0.29<\/li>\r\n<\/ul>\r\nThen, the predicted values (i.e., the regression line on Figure 10.3 above) are:\r\n\r\n&nbsp;\r\n\r\n$\\textrm{predicted years of schooling}=\\hat{y}=a+bx=10.67+0.29(\\textrm{years of parental schooling})$\r\n\r\n&nbsp;\r\n\r\nWe also know that the standard error of<em> b<\/em> is <em>s<sub>b<\/sub><\/em>=0.016, so\r\n\r\n&nbsp;\r\n\r\n$$t=\\frac{b}{s_b}=\\frac{0.29}{0.016}=18.607$$ [footnote]If you actually divide 0.29 by 0.016, you wil end up with 18.125. The difference from 18.607 is due to rounding (as the standard error of <em>b<\/em> is rounded up to 0.016 from 0.01558...).[\/footnote]\r\n\r\n&nbsp;\r\n\r\n<strong>Thus, with <em>t<\/em>=18.607, <em>df<\/em>=1,684, and <em>p<\/em>&lt;<em>\u03b1<\/em>=0.001, we can reject the null hypothesis. Our current evidence supports our hypothesis that father's education affects their offspring's education, on average. The effect is 0.29 years (or about 3.5 months) for every additional year of father's schooling, and it is statistically significant.<\/strong>\r\n\r\n&nbsp;\r\n\r\nAs well, we can interpret <strong>the confidence interval:<\/strong>\r\n<ul>\r\n \t<li>95% CI: $$ b\\pm1.96s_b =0.29\\pm1.96(0.016)=0.29\\pm0.031=(0.26; 0.32)$$<\/li>\r\n<\/ul>\r\n<strong>Or, father's education's effect on offspring's education would be between 0.26 additional years and 0.32 additional years for every year of father's schooling with 95% certainty; in other words, the effect would be 0.29\u00a0\u00b1 0.031, 19 out of 20 times.<\/strong>\r\n\r\n&nbsp;\r\n\r\nThat's a lot more information than simply stating that the variables are associated based on the correlation coefficient!\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\nNow let's make sure you understand how regression works and where the regression coefficients and line come from by interpreting regression output.\r\n\r\n&nbsp;\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\"><em>Do It! 10.2\u00a0 Class Attendance and Final Test Scores (Simulated Data)<\/em><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n&nbsp;\r\n\r\nWe are revisiting the simulated data on student class attendance (measured in percent of classes attended) and their final class scores. <em>N<\/em>=987.\u00a0<span style=\"font-size: 1rem\">Start by stating your hypotheses, then, using the\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">SPSS's output presented in Figure 10.4 and Table 10.5 below, write a paragraph interpreting what you have found, discussing the evidence presented regarding your hypotheses and your decision about them, etc. Include as much information as possible, and do not forget to justify your use of linear regression in this case.<\/span>\r\n\r\n&nbsp;\r\n\r\n<em>Figure 10.4 Class Attendance and Final Test Scores<\/em>\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/scatterplot-attendance-scores-full.png\" alt=\"\" width=\"462\" height=\"370\" class=\"wp-image-1377 size-full aligncenter\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<em>Table 10.5 Class Attendance and Final Test Scores<\/em>\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/regression-attendance-scores-full.png\" alt=\"\" width=\"733\" height=\"163\" class=\"wp-image-1378 size-full aligncenter\" \/>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\nFinally, these are the steps through which the regression output is obtained in SPSS.\r\n\r\n&nbsp;\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\"><em>SPSS Tip 10.1 Linear Regression<\/em><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>From the <em>Main Menu<\/em>, select <em>Analyze<\/em>, then from the pull-down menu, select <em>Regression<\/em> and click on <em>Linear<\/em>;<\/li>\r\n \t<li>Select your dependent variable from the list of variables on the left and, using the appropriate arrow, move it to the <em>Dependent<\/em> open space on the right;<\/li>\r\n \t<li>Select your independent variable from the list of variables on the left and, using the appropriate arrow, move it to the <em>Block 1 of 1<\/em> empty space on the right.<\/li>\r\n \t<li>You can click <em>OK<\/em> or, if you need a confidence interval for <em>b<\/em>, click on <em>Statistics<\/em>, and check off <em>Confidence intervals<\/em> in the new window (here you can also specify the confidence <em>Level<\/em> of the CI); click <em>Continue<\/em>;<\/li>\r\n \t<li>Once back in the original window, click <em>OK<\/em>.<\/li>\r\n \t<li>After the <em>OK<\/em>, SPSS will provide the output in the<em> Output<\/em> window. The relevant information we have discussed so far can be found in the last table called <em>Coefficients<\/em>.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\nSPSS provides several tables as the standard regression output. Beyond the <em>Coefficients<\/em> one, there are three other short tables: a <em>Variables Entered\/Removed<\/em> (which lists the independent variable\/s in the model and the dependent variable as a footnote), an <em>ANOVA<\/em> table (which presents analysis of variance information that, as mentioned before, is outside the scope of this book), and a <em>Model Summary<\/em> table. We qill take a brief look at that last table in the next section.","rendered":"<p>To test the regression coefficient\u00a0<em>b<\/em> for significance we have the following hypotheses:<\/p>\n<ul>\n<li>H<sub>0<\/sub>: The independent variable <em>x<\/em> has no effect on the dependent variable <em>y<\/em> (i.e., the variables are not associated); <em>\u03b2<\/em>=0.<\/li>\n<li>H<sub>a<\/sub>: The independent variable<em> x<\/em> has an effect on the dependent variable<em> y<\/em> (i.e., the variables are associated);\u00a0<em>\u03b2<\/em>\u22600<a class=\"footnote\" title=\"Note that I am using causal language here with the assumption that the conditions for causality are met. Theirs is a separate investigation. In and of itself, finding a significant effect of x on y does not itself establish that changes in x cause\u00a0changes in\u00a0y.\" id=\"return-footnote-2151-1\" href=\"#footnote-2151-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><strong>After calculating<em> t<sub>b<\/sub><\/em> with <em>df<\/em>=<em>N<\/em>-2 and finding its associated <em>p<\/em>-value, we then compare the <em>p<\/em>-value to the pre-selected significance level\u00a0<em>\u03b1<\/em>. As usual, when<em> p<\/em>\u2264<em>\u03b1<\/em>, we reject the null hypothesis, and have enough evidence to deem the regression coefficient <em>b<\/em> statistically significant. If, on the contrary, <em>p<\/em>&gt;<em>\u03b1<\/em>, we fail to reject the null hypothesis and therefore conclude that at present there is no evidence to suggest an effect of <em>x<\/em> on <em>y<\/em>.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Again, similarly to other statistics, we can <strong>calculate confidence intervals for <em>b<\/em>, so that we can report the size of the effect with a specific level of certainty<\/strong>. For example, the 95% confidence interval for the regression coefficient <em>b<\/em> is:<\/p>\n<ul>\n<li>95% CI: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/ql-cache\/quicklatex.com-b6e9df8524c7a563967d70aec00fbc8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#112;&#109;&#32;&#49;&#46;&#57;&#54;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#115;&#95;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>To illustrate, let&#8217;s revisit our example about the effect of parental education on their offspring education. (Don&#8217;t worry,\u00a0<span style=\"text-indent: 18.6667px;font-size: 14pt\">with <em>N<\/em>=1, 686\u00a0<\/span><span style=\"text-indent: 1em;font-size: 14pt\">I will not offer you a calculation by hand: SPSS is there for us.)\u00a0\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\"><em>Example 10.4 Effect of Parental Years of Schooling on Respondent&#8217;s Years of Schooling (GSS 2018)<\/em><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>&nbsp;<\/p>\n<p>We already examined the association between parental and offspring education through the correlation coefficient <em>r<\/em> and found it to be moderately weak at 0.413, and statistically significant at\u00a0<em>\u03b1<\/em>=0.01. Can we do better, however, and estimate the effect of each additional year of parental schooling on the schooling of the respondents?<\/p>\n<p>&nbsp;<\/p>\n<p>Again, we use data from the U.S.\u00a0<em>GSS 2018<\/em> (NORC, 2019).\u00a0 Our sample is <em>N<\/em>=1,686, and <strong>our hypotheses are:<\/strong><\/p>\n<ul>\n<li>H<sub>0<\/sub>: Father&#8217;s education has no effect on respondent&#8217;s education;\u00a0<em>\u03b2<\/em>=0.<\/li>\n<li>H<sub>a<\/sub>: Father&#8217;s education has an effect on respondent&#8217;s education;\u00a0<em>\u03b2<\/em>\u22600.<\/li>\n<\/ul>\n<p><strong>The regression model is:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/ql-cache\/quicklatex.com-b1b1043436d90ec88e574eeaa0b462f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#114;&#109;&#123;&#121;&#101;&#97;&#114;&#115;&#32;&#111;&#102;&#32;&#115;&#99;&#104;&#111;&#111;&#108;&#105;&#110;&#103;&#125;&#61;&#121;&#61;&#97;&#43;&#98;&#120;&#43;&#101;&#61;&#97;&#43;&#98;&#40;&#92;&#116;&#101;&#120;&#116;&#114;&#109;&#123;&#121;&#101;&#97;&#114;&#115;&#32;&#111;&#102;&#32;&#112;&#97;&#114;&#101;&#110;&#116;&#97;&#108;&#32;&#115;&#99;&#104;&#111;&#111;&#108;&#105;&#110;&#103;&#125;&#41;&#43;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"584\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Our predicted values are:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/ql-cache\/quicklatex.com-315df1e2b4ab603a552729503ae92719_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#114;&#109;&#123;&#112;&#114;&#101;&#100;&#105;&#99;&#116;&#101;&#100;&#32;&#121;&#101;&#97;&#114;&#115;&#32;&#111;&#102;&#32;&#115;&#99;&#104;&#111;&#111;&#108;&#105;&#110;&#103;&#125;&#32;&#61;&#92;&#104;&#97;&#116;&#123;&#121;&#125;&#61;&#97;&#43;&#98;&#120;&#61;&#97;&#43;&#98;&#40;&#92;&#116;&#101;&#120;&#116;&#114;&#109;&#123;&#121;&#101;&#97;&#114;&#115;&#32;&#111;&#102;&#32;&#112;&#97;&#114;&#101;&#110;&#116;&#97;&#108;&#32;&#115;&#99;&#104;&#111;&#111;&#108;&#105;&#110;&#103;&#125;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"582\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Figure 10.3 plots the association and Table 10.4 show the relevant SPSS output.<\/p>\n<p>&nbsp;<\/p>\n<p>Figure 10.3\u00a0<em>Linear Regression of Respondent&#8217;s Years of Schooling and Father&#8217;s Years of Schooling<\/em><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/scatterplot-educ-paeduc-line.png\" alt=\"\" width=\"462\" height=\"370\" class=\"wp-image-1371 size-full aligncenter\" srcset=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/scatterplot-educ-paeduc-line.png 462w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/scatterplot-educ-paeduc-line-300x240.png 300w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/scatterplot-educ-paeduc-line-65x52.png 65w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/scatterplot-educ-paeduc-line-225x180.png 225w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/scatterplot-educ-paeduc-line-350x280.png 350w\" sizes=\"auto, (max-width: 462px) 100vw, 462px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><em>Table 10.4 Linear Regression of Respondent&#8217;s Years of Schooling and Father&#8217;s Years of Schooling<\/em><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/regression-table-educ-paeduc.png\" alt=\"\" width=\"812\" height=\"178\" class=\"wp-image-1370 size-full aligncenter\" srcset=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/regression-table-educ-paeduc.png 812w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/regression-table-educ-paeduc-300x66.png 300w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/regression-table-educ-paeduc-768x168.png 768w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/regression-table-educ-paeduc-65x14.png 65w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/regression-table-educ-paeduc-225x49.png 225w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/04\/regression-table-educ-paeduc-350x77.png 350w\" sizes=\"auto, (max-width: 812px) 100vw, 812px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>That is, SPSS has calculated the constant (or <em>Y<\/em>-intercept) <em>a<\/em> and the regression coefficient <em>b<\/em> in such a way as to minimize the residuals:<\/p>\n<ul>\n<li><em>a<\/em> = 10.67<\/li>\n<li><em>b<\/em> = 0.29<\/li>\n<\/ul>\n<p>Then, the predicted values (i.e., the regression line on Figure 10.3 above) are:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/ql-cache\/quicklatex.com-499ad3fbf95584e85fe972c04fd1a891_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#114;&#109;&#123;&#112;&#114;&#101;&#100;&#105;&#99;&#116;&#101;&#100;&#32;&#121;&#101;&#97;&#114;&#115;&#32;&#111;&#102;&#32;&#115;&#99;&#104;&#111;&#111;&#108;&#105;&#110;&#103;&#125;&#61;&#92;&#104;&#97;&#116;&#123;&#121;&#125;&#61;&#97;&#43;&#98;&#120;&#61;&#49;&#48;&#46;&#54;&#55;&#43;&#48;&#46;&#50;&#57;&#40;&#92;&#116;&#101;&#120;&#116;&#114;&#109;&#123;&#121;&#101;&#97;&#114;&#115;&#32;&#111;&#102;&#32;&#112;&#97;&#114;&#101;&#110;&#116;&#97;&#108;&#32;&#115;&#99;&#104;&#111;&#111;&#108;&#105;&#110;&#103;&#125;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"582\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>We also know that the standard error of<em> b<\/em> is <em>s<sub>b<\/sub><\/em>=0.016, so<\/p>\n<p>&nbsp;<\/p>\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 40px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/ql-cache\/quicklatex.com-1d1f8cbd22fff3208117b67bb03cd621_l3.png\" height=\"40\" width=\"190\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#091;&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#115;&#95;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#48;&#46;&#50;&#57;&#125;&#123;&#48;&#46;&#48;&#49;&#54;&#125;&#61;&#49;&#56;&#46;&#54;&#48;&#55;&#92;&#093;\" title=\"Rendered by QuickLaTeX.com\" \/><\/p>\n<p> <a class=\"footnote\" title=\"If you actually divide 0.29 by 0.016, you wil end up with 18.125. The difference from 18.607 is due to rounding (as the standard error of b is rounded up to 0.016 from 0.01558...).\" id=\"return-footnote-2151-2\" href=\"#footnote-2151-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Thus, with <em>t<\/em>=18.607, <em>df<\/em>=1,684, and <em>p<\/em>&lt;<em>\u03b1<\/em>=0.001, we can reject the null hypothesis. Our current evidence supports our hypothesis that father&#8217;s education affects their offspring&#8217;s education, on average. The effect is 0.29 years (or about 3.5 months) for every additional year of father&#8217;s schooling, and it is statistically significant.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>As well, we can interpret <strong>the confidence interval:<\/strong><\/p>\n<ul>\n<li>95% CI:\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 18px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/ql-cache\/quicklatex.com-184ac451d0da4868a9242873f576722f_l3.png\" height=\"18\" width=\"463\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#091;&#98;&#92;&#112;&#109;&#49;&#46;&#57;&#54;&#115;&#95;&#98;&#32;&#61;&#48;&#46;&#50;&#57;&#92;&#112;&#109;&#49;&#46;&#57;&#54;&#40;&#48;&#46;&#48;&#49;&#54;&#41;&#61;&#48;&#46;&#50;&#57;&#92;&#112;&#109;&#48;&#46;&#48;&#51;&#49;&#61;&#40;&#48;&#46;&#50;&#54;&#59;&#32;&#48;&#46;&#51;&#50;&#41;&#92;&#093;\" title=\"Rendered by QuickLaTeX.com\" \/><\/p>\n<\/li>\n<\/ul>\n<p><strong>Or, father&#8217;s education&#8217;s effect on offspring&#8217;s education would be between 0.26 additional years and 0.32 additional years for every year of father&#8217;s schooling with 95% certainty; in other words, the effect would be 0.29\u00a0\u00b1 0.031, 19 out of 20 times.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>That&#8217;s a lot more information than simply stating that the variables are associated based on the correlation coefficient!<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Now let&#8217;s make sure you understand how regression works and where the regression coefficients and line come from by interpreting regression output.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\"><em>Do It! 10.2\u00a0 Class Attendance and Final Test Scores (Simulated Data)<\/em><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>&nbsp;<\/p>\n<p>We are revisiting the simulated data on student class attendance (measured in percent of classes attended) and their final class scores. <em>N<\/em>=987.\u00a0<span style=\"font-size: 1rem\">Start by stating your hypotheses, then, using the\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">SPSS&#8217;s output presented in Figure 10.4 and Table 10.5 below, write a paragraph interpreting what you have found, discussing the evidence presented regarding your hypotheses and your decision about them, etc. Include as much information as possible, and do not forget to justify your use of linear regression in this case.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><em>Figure 10.4 Class Attendance and Final Test Scores<\/em><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/scatterplot-attendance-scores-full.png\" alt=\"\" width=\"462\" height=\"370\" class=\"wp-image-1377 size-full aligncenter\" srcset=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/scatterplot-attendance-scores-full.png 462w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/scatterplot-attendance-scores-full-300x240.png 300w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/scatterplot-attendance-scores-full-65x52.png 65w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/scatterplot-attendance-scores-full-225x180.png 225w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/scatterplot-attendance-scores-full-350x280.png 350w\" sizes=\"auto, (max-width: 462px) 100vw, 462px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><em>Table 10.5 Class Attendance and Final Test Scores<\/em><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/regression-attendance-scores-full.png\" alt=\"\" width=\"733\" height=\"163\" class=\"wp-image-1378 size-full aligncenter\" srcset=\"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/regression-attendance-scores-full.png 733w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/regression-attendance-scores-full-300x67.png 300w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/regression-attendance-scores-full-65x14.png 65w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/regression-attendance-scores-full-225x50.png 225w, https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-content\/uploads\/sites\/564\/2019\/05\/regression-attendance-scores-full-350x78.png 350w\" sizes=\"auto, (max-width: 733px) 100vw, 733px\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Finally, these are the steps through which the regression output is obtained in SPSS.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\"><em>SPSS Tip 10.1 Linear Regression<\/em><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>From the <em>Main Menu<\/em>, select <em>Analyze<\/em>, then from the pull-down menu, select <em>Regression<\/em> and click on <em>Linear<\/em>;<\/li>\n<li>Select your dependent variable from the list of variables on the left and, using the appropriate arrow, move it to the <em>Dependent<\/em> open space on the right;<\/li>\n<li>Select your independent variable from the list of variables on the left and, using the appropriate arrow, move it to the <em>Block 1 of 1<\/em> empty space on the right.<\/li>\n<li>You can click <em>OK<\/em> or, if you need a confidence interval for <em>b<\/em>, click on <em>Statistics<\/em>, and check off <em>Confidence intervals<\/em> in the new window (here you can also specify the confidence <em>Level<\/em> of the CI); click <em>Continue<\/em>;<\/li>\n<li>Once back in the original window, click <em>OK<\/em>.<\/li>\n<li>After the <em>OK<\/em>, SPSS will provide the output in the<em> Output<\/em> window. The relevant information we have discussed so far can be found in the last table called <em>Coefficients<\/em>.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>SPSS provides several tables as the standard regression output. Beyond the <em>Coefficients<\/em> one, there are three other short tables: a <em>Variables Entered\/Removed<\/em> (which lists the independent variable\/s in the model and the dependent variable as a footnote), an <em>ANOVA<\/em> table (which presents analysis of variance information that, as mentioned before, is outside the scope of this book), and a <em>Model Summary<\/em> table. We qill take a brief look at that last table in the next section.<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-2151-1\">Note that I am using causal language here with the assumption that the conditions for causality are met. Theirs is a separate investigation. In and of itself, finding a significant effect of <em>x<\/em> on <em>y<\/em> does not itself establish that changes in<em> x<\/em> <em>cause<\/em>\u00a0changes in\u00a0<em>y<\/em>. <a href=\"#return-footnote-2151-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-2151-2\">If you actually divide 0.29 by 0.016, you wil end up with 18.125. The difference from 18.607 is due to rounding (as the standard error of <em>b<\/em> is rounded up to 0.016 from 0.01558...). <a href=\"#return-footnote-2151-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":533,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2151","chapter","type-chapter","status-publish","hentry"],"part":128,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/pressbooks\/v2\/chapters\/2151","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/wp\/v2\/users\/533"}],"version-history":[{"count":10,"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/pressbooks\/v2\/chapters\/2151\/revisions"}],"predecessor-version":[{"id":2163,"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/pressbooks\/v2\/chapters\/2151\/revisions\/2163"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/pressbooks\/v2\/parts\/128"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/pressbooks\/v2\/chapters\/2151\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/wp\/v2\/media?parent=2151"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/pressbooks\/v2\/chapter-type?post=2151"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/wp\/v2\/contributor?post=2151"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/simplestats\/wp-json\/wp\/v2\/license?post=2151"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}