{"id":4961,"date":"2015-08-19T13:21:13","date_gmt":"2015-08-19T18:21:13","guid":{"rendered":"http:\/\/www.ssc.wisc.edu\/~jfrees\/?page_id=4961"},"modified":"2015-08-20T12:15:54","modified_gmt":"2015-08-20T17:15:54","slug":"example-term-life-insurance-continued","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/chapter-4-multiple-linear-regression-ii\/4-1-the-role-of-binary-variables\/example-term-life-insurance-continued\/","title":{"rendered":"Example: Term Life Insurance &#8211; Continued"},"content":{"rendered":"<p>We now return to the marital status of respondents from the Survey of Consumer Finances (SCF). Recall that marital status is not measured continuously but rather takes on values that falls into distinct groups that we treat as unordered. In <a style=\"font-size: 14px\" href='http:\/\/www.ssc.wisc.edu\/~jfrees\/regression\/chapter-3-multiple-linear-regression\/'>Chapter 3<\/a>, we grouped survey respondents according to whether or not they are &#8220;single,&#8221; where being single includes never married, separated, divorced, widowed, and are not married and living with a partner. We now supplement this by considering the categorical variable, <em>MARSTAT<\/em>, that represents the marital status of the survey respondent. This may be: <\/p>\n<ul>\n<li> 1, for married  <\/li>\n<li> 2, for living with partner  <\/li>\n<li> 0, for other (SCF further breaks down this category into  separated, divorced, widowed, never married and inapplicable,  persons age 17 or less, no further persons).  <\/li>\n<\/ul>\n<p> As before, the dependent variable is <em>y = LNFACE<\/em>, the amount that the company will pay in the event of the death of the named insured (in logarithmic dollars). Table 4.1 summarizes the dependent variable by level of the categorial variable. This table shows that the marital status &#8220;married&#8221; is the most prevalent in the sample and that those married choose to have the most life insurance coverage. Figure 4.1 gives a more complete picture of the distribution of <em>LNFACE<\/em> for each of the three types of marital status. The table and figure also suggests that those living together have less life insurance coverage than the other two categories.<br \/>\n\\begin{matrix}\\begin{array}{c}<br \/>\n\\text{Table 4.1 Summary Statistics of Logarithmic Face By Marital Status}<br \/>\n\\end{array}\\\\\\small<br \/>\n\\begin{array}{lcccc} \\hline &#038;  &#038;  &#038;  &#038; \\text{Standard} \\\\ &#038; \\text{MARSTAT} &#038; \\text{Number} &#038; \\text{Mean} &#038; \\text{deviation}\\\\\\hline \\text{Other}           &#038; 0 &#038; 57 &#038; 10.958 &#038; 1.566 \\\\ \\text{Married}         &#038; 1 &#038; 208 &#038; 12.329 &#038; 1.822 \\\\ \\text{Living together} &#038; 2 &#038; 10 &#038; 10.825 &#038; 2.001 \\\\ \\hline \\text{Total}           &#038;   &#038; 275 &#038; 11.990 &#038; 1.871 \\\\  \\hline \\end{array} \\end{matrix} <\/p>\n<h2 style=\"text-align: center;\"><a id=\"displayTextt1\" href=\"javascript:togglecode('toggleTextt1','displayTextt1');\"><i><strong>R Code for Table 4.1<\/strong><\/i><\/a> <\/h2>\n<div id=\"toggleTextt1\" style=\"display: none\">\n<pre><strong>R-Code<\/strong>\r\nTerm &lt;- read.table('http:\/\/instruction.bus.wisc.edu\/jfrees\/jfreesbooks\/Regression%20Modeling\/BookWebDec2010\/CSVData\/TermLife.csv', header=TRUE, sep=\",\")\r\n\r\n#  PICK THE SUBSET OF THE DATA CORRESPONDING TO TERM PURCHASE\r\nTerm2 &lt;- subset(Term, subset=FACE > 0)\r\nnames(Term2)\r\n<\/pre>\n<pre><strong>R-Code Output<\/strong>\r\n [1] \"GENDER\"             \"AGE\"                \"MARSTAT\"            \"EDUCATION\"          \"ETHNICITY\"         \r\n [6] \"SMARSTAT\"           \"SGENDER\"            \"SAGE\"               \"SEDUCATION\"         \"NUMHH\"             \r\n[11] \"INCOME\"             \"TOTINCOME\"          \"CHARITY\"            \"FACE\"               \"FACECVLIFEPOLICIES\"\r\n[16] \"CASHCVLIFEPOLICIES\" \"BORROWCVLIFEPOL\"    \"NETVALUE\"          \r\n<\/pre>\n<pre><strong>R-Code<\/strong>\r\nattach(Term2)\r\nTerm2$LNFACE &lt;- with(Term2, log(FACE))\r\nTerm2$LNINCOME &lt;- with (Term2, log(INCOME))\r\nTerm2$MARSTAT &lt;- as.factor(Term2$MARSTAT)\r\ntable(Term2$MARSTAT)\r\n<\/pre>\n<pre><strong>R-Code Output<\/strong>\r\n  0   1   2 \r\n 57 208  10\r\n<\/pre>\n<pre><strong>R-Code<\/strong>\r\n#  SUMMARY BY LEVEL OF MARSTAT\r\nlibrary(Rcmdr)\r\nlibrary(abind)\r\nnumSummary(Term2[, \"LNFACE\"], groups=Term2$MARSTAT, statistics=c(\"mean\", \"sd\"))\r\n<\/pre>\n<pre><strong>R-Code Output<\/strong>\r\n      mean       sd data:n\r\n0 10.95842 1.566224     57\r\n1 12.32909 1.822243    208\r\n2 10.82507 2.000644     10\r\n<\/pre>\n<pre><strong>R-Code<\/strong>\r\nnumSummary(Term2[, \"LNFACE\"], statistics=c(\"mean\", \"sd\"))\r\n<\/pre>\n<pre><strong>R-Code Output<\/strong>\r\n     mean       sd   n\r\n 11.99029 1.870728 275\r\n<\/pre>\n<\/div>\n<figure class=\"wp-caption aligncenter\" style=\"max-width: 300px;\" aria-label=\"Figure 4.1 Box Plots of Logarithmic Face, by Level of Marital Status\"><a href=\"http:\/\/www.ssc.wisc.edu\/~jfrees\/wp-content\/uploads\/2015\/04\/F4BoxFACEMARSTAT.png\"><img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.ssc.wisc.edu\/~jfrees\/wp-content\/uploads\/2015\/04\/F4BoxFACEMARSTAT.png\" alt=\"F4BoxFACEMARSTAT\" width=\"432\" height=\"288\" class=\"aligncenter size-full wp-image-3268\" srcset=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-content\/uploads\/2015\/04\/F4BoxFACEMARSTAT.png 432w, https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-content\/uploads\/2015\/04\/F4BoxFACEMARSTAT-300x200.png 300w\" sizes=\"(max-width: 432px) 100vw, 432px\" \/><\/a><figcaption class=\"wp-caption-text\">Figure 4.1 Box Plots of Logarithmic Face, by Level of Marital Status<\/figcaption><\/figure>\n<h2 style=\"text-align: center;\"><a id=\"displayTextf1\" href=\"javascript:togglecode('toggleTextf1','displayTextf1');\"><i><strong>R Code for Figure 4.1<\/strong><\/i><\/a> <\/h2>\n<div id=\"toggleTextf1\" style=\"display: none\">\n<pre><strong>R-Code<\/strong>\r\nboxplot(LNFACE ~ MARSTAT, ylab=\"LNFACE\", xlab=\"MARSTAT\", data=Term2)\r\n<\/pre>\n<\/div>\n<p> Are the continuous and categorical variables jointly important determinants of response? To answer this, a regression was run using <em>LNFACE<\/em> as the response and five explanatory variables, three continuous and two binary (for marital status). Recall that our three continuous explanatory variables are:  <em>LNINCOME<\/em> (logarithmic annual income), the number of years of EDUCATION of the survey respondent and the number of household members, <em>NUMHH<\/em>. <\/p>\n<p> For the binary variables, first define <em>MAR<\/em>0 to be the binary variable that is one if <em>MARSTAT<\/em>=0 and zero otherwise. Similarly, define <em>MAR<\/em>1 and <em>MAR<\/em>2 to be binary variables that indicate <em>MARSTAT<\/em>=1 and <em>MARSTAT<\/em>=2, respectively. There is a perfect linear dependency among these three binary variables in that <em>MAR<\/em>0 + <em>MAR<\/em>1 + <em>MAR<\/em>2 = 1 for any survey respondent. Thus, we need only two of the three. However, there is <em>not<\/em> a perfect dependency among any two of the three. It turns out that Corr(MAR0,MAR1) = -0.90, Corr(MAR0,MAR2) =-0.10 and Corr(MAR1,MAR2) = -0.34. <\/p>\n<h2 style=\"text-align: center;\"><a id=\"displayText\" href=\"javascript:togglecode('toggleText','displayText');\"><i><strong>R Code to Compute Correlation<\/strong><\/i><\/a> <\/h2>\n<div id=\"toggleText\" style=\"display: none\">\n<pre><strong>R-Code<\/strong>\r\n#  MAKE BINARY VARIABLES\r\nTerm2$MAR0 &lt;- with(Term2, 1*(MARSTAT == 0))\r\nTerm2$MAR1 &lt;- with(Term2, 1*(MARSTAT == 1))\r\nTerm2$MAR2 &lt;- with(Term2, 1*(MARSTAT == 2))\r\n\r\nCheck1 &lt;- data.frame(MARSTAT, Term2$MAR0, Term2$MAR1, Term2$MAR2)\r\n#fix(Check1)\r\n\r\n#  CHECK THE DEPENDENCIES AMONG MARO, MAR1, MAR2\r\ncor(Term2[, c(\"MAR0\", \"MAR1\", \"MAR2\")], use=\"complete.obs\")\r\n<\/pre>\n<pre><strong>R-Code Output<\/strong>\r\n            MAR0       MAR1        MAR2\r\nMAR0  1.00000000 -0.9009557 -0.09933133\r\nMAR1 -0.90095572  1.0000000 -0.34227197\r\nMAR2 -0.09933133 -0.3422720  1.00000000\r\n<\/pre>\n<\/div>\n<p> A regression model was run using <em>LNINCOME<\/em>, <em>EDUCATION<\/em>, <em>NUMHH<\/em>, <em>MAR<\/em>0 and <em>MAR<\/em>2 as explanatory variables. The fitted regression equation turns out to be \\begin{eqnarray*} \\widehat{y} &#038;=&#038; 2.605 + 0.452 \\textrm{LNINCOME} +0.205 \\textrm{EDUCATION} + 0.248 \\textrm{NUMHH} \\\\  &#038; &#038; ~~ -0.557 \\textrm{MAR0} -0.789 \\textrm{MAR2}. \\end{eqnarray*} <\/p>\n<h2 style=\"text-align: center;\"><a id=\"displayTextr1\" href=\"javascript:togglecode('toggleTextr1','displayTextr1');\"><i><strong>R Code for Regression<\/strong><\/i><\/a> <\/h2>\n<div id=\"toggleTextr1\" style=\"display: none\">\n<pre><strong>R-Code<\/strong>\r\nsummary(lm(LNFACE ~ LNINCOME+EDUCATION+NUMHH +MAR0+MAR2, data=Term2))\r\n<\/pre>\n<pre><strong>R-Code Output<\/strong>\r\nCall:\r\nlm(formula = LNFACE ~ LNINCOME + EDUCATION + NUMHH + MAR0 + MAR2, \r\n    data = Term2)\r\n\r\nResiduals:\r\n    Min      1Q  Median      3Q     Max \r\n-5.8875 -0.8505  0.1124  0.8468  4.5173 \r\n\r\nCoefficients:\r\n            Estimate Std. Error t value Pr(>|t|)    \r\n(Intercept)  3.39477    0.90019   3.771 0.000200 ***\r\nLNINCOME     0.45151    0.07872   5.736 2.61e-08 ***\r\nEDUCATION    0.20467    0.03862   5.299 2.42e-07 ***\r\nNUMHH        0.24770    0.06940   3.569 0.000424 ***\r\nMAR0        -0.55707    0.25929  -2.148 0.032574 *  \r\nMAR2        -0.78941    0.49532  -1.594 0.112169    \r\n---\r\nSignif. codes:  0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1\r\n\r\nResidual standard error: 1.513 on 269 degrees of freedom\r\nMultiple R-squared:  0.358,\tAdjusted R-squared:  0.3461 \r\nF-statistic:    30 on 5 and 269 DF,  p-value: < 2.2e-16\r\n<\/pre>\n<\/div>\n<p>To interpret the regression coefficients associated with marital status, consider a respondent who is married. In this case, then <em>MAR<\/em>0=0, <em>MAR<\/em>1=1 and<em> MAR<\/em>2=0, so that \\begin{eqnarray*} \\widehat{y}_m &=& 2.605 + 0.452 \\textrm{LNINCOME} +0.205 \\textrm{EDUCATION} + 0.248 \\textrm{NUMHH} . \\end{eqnarray*} Similarly, if the respondent is coded as living together, then <em>MAR<\/em>0=0, <em>MAR<\/em>1=0 and <em>MAR<\/em>2=1, and<br \/>\n\\begin{align} \\widehat{y}_{lt} &= 2.605 + 0.452 \\textrm{LNINCOME} +0.205 \\textrm{EDUCATION} + 0.248 \\textrm{NUMHH}\\\\ &-0.789. \\end{align} The difference between \\(\\widehat{y}_m\\) and \\(\\widehat{y}_{lt}\\) is \\(0.789.\\) Thus, we may interpret the regression coefficient associated with <em>MAR<\/em>2, -0.789, to be the difference in fitted values for someone living together compared to a similar person who is married (the omitted category). <\/p>\n<p> Similarly, we can interpret -0.557 to be the difference between the ``other'' category and the married category, holding other explanatory variables fixed. For the difference in fitted values between the ``other'' and the ``living together'' categories, we may use \\(-0.557 - (-0.789) = 0.232.\\) <\/p>\n<p> Although the regression was run using <em>MAR<\/em>0 and <em>MAR<\/em>2, any two out of the three would produce the same ANOVA Table 4.2. However, the choice of binary variables does impact the regression coefficients. Table 4.3 shows three models, omitting <em>MAR<\/em>1, <em>MAR<\/em>2 and <em>MAR<\/em>0, respectively. For each fit, the coefficients associated with the continuous variables remain the same. As we have seen, the binary variable interpretations are with respect to the omitted category, known as the <em>reference level<\/em>. Although they change from model to model, they overall interpretation remains the same. That is, if we would like to estimate the difference in coverage between the ``other'' and the ``living together'' category, the estimate would be 0.232, regardless of the model. <\/p>\n<p> \\begin{matrix}\\begin{array}{c}<br \/>\n\\text{Table 4.2 Term Life with Marital Status ANOVA Table}<br \/>\n\\end{array}\\\\\\small<br \/>\n\\begin{array}{lrrr}  \\hline \\text{Source} & \\text{Sum of Squares} & df & \\text{Mean Square} \\\\ \\hline<br \/>\n \\text{Regression} & 343.28 & 5 &   68.66 \\\\ \\text{Error}      & 615.62 & 269 &  2.29 \\\\ \\text{Total}      & 948.90& 274 &   \\\\ \\hline<br \/>\n\\end{array}\\\\\\scriptsize<br \/>\n\\begin{array}{l}<br \/>\n\\text{Residual Standard Error} s= 1.513, R^2 = 35.8\\%, R_a^2 = 34.6\\%\\end{array}  \\end{matrix} <\/p>\n<h2 style=\"text-align: center;\"><a id=\"displayTextt2\" href=\"javascript:togglecode('toggleTextt2','displayTextt2');\"><i><strong>R Code for Table 4.2<\/strong><\/i><\/a> <\/h2>\n<div id=\"toggleTextt2\" style=\"display: none\">\n<pre><strong>R-Code<\/strong>\r\nanova(lm(LNFACE ~ LNINCOME+EDUCATION+NUMHH +MAR0+MAR2, data=Term2))\r\n<\/pre>\n<pre><strong>R-Code Output<\/strong>\r\nAnalysis of Variance Table\r\n\r\nResponse: LNFACE\r\n           Df Sum Sq Mean Sq F value    Pr(>F)    \r\nLNINCOME    1 222.63 222.629 97.2800 < 2.2e-16 ***\r\nEDUCATION   1  51.50  51.502 22.5044 3.407e-06 ***\r\nNUMHH       1  54.34  54.336 23.7426 1.883e-06 ***\r\nMAR0        1   9.00   8.999  3.9321   0.04839 *  \r\nMAR2        1   5.81   5.813  2.5400   0.11217    \r\nResiduals 269 615.62   2.289                      \r\n---\r\nSignif. codes:  0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1\r\n<\/pre>\n<\/div>\n<p> Although the three models in Table 4.3 are the same except for different choices of parameters, they do appear different. In particular, the \\(t\\)-ratios differ and give different appearances of statistical significance. For example, both of the \\(t\\)-ratios associated with marital status in Model 2 are less than 2 in absolute value, suggesting that marital status is unimportant. In contrast, both Models 1 and 3 have at least one marital status binary that exceeds 2 in absolute value, suggesting statistical significance. Thus, you can influence the <em>appearance<\/em> of statistical significance by altering the choice of the reference level. To assess the overall importance of marital status (not just each binary variable), <a  style=\"font-size: 14px\" href='http:\/\/www.ssc.wisc.edu\/~jfrees\/regression\/chapter-4-multiple-linear-regression-ii\/4-2-statistical-inference-for-several-coefficients\/'><br \/>\nSection 4.2<\/a> will introduce tests of sets of regression coefficients. <\/p>\n<p>  \\begin{matrix}\\begin{array}{c}<br \/>\n\\text{Table 4.3 Term Life Regression Coefficients with Marital Status}<br \/>\n\\end{array}\\\\\\scriptsize<br \/>\n\\begin{array}{llll}<br \/>\n\\hline \\phantom{XXXXXXXXXXX} & \\text{Model 1}\\phantom{XXXXX}& \\phantom{XX}\\text{Model 2}\\phantom{XXXXX}& \\phantom{XX}\\text{Model 3}\\phantom{XXXXX}\\\\<br \/>\n\\end{array}\\\\\\scriptsize<br \/>\n\\begin{array}{l|rr|rr|rr}   \\hline  \\text{Explanatory} \\\\  \\text{Variable} & \\text{Coefficient} & t-\\text{ratio} & \\text{Coefficient} & t-\\text{ratio}& \\text{Coefficient} &  t-\\text{ratio}\\\\\\hline \\text{LNINCOME} & 0.452 & 5.74 & 0.452 & 5.74 & 0.452 & 5.74 \\\\ \\text{EDUCATION} &0.205 & 5.30 &0.205 & 5.30&0.205 & 5.30 \\\\ \\text{NUMHH}     & 0.248 & 3.57 & 0.248 & 3.57 & 0.248 & 3.57 \\\\\\hline \\text{Intercept} & 3.395 & 3.77  & 2.605&  2.74 & 2.838 & 3.34\\\\ \\text{MAR0}    & -0.557 & -2.15&  0.232 &  0.44\\\\ \\text{MAR1} & & & 0.789 & 1.59 & 0.557 & 2.15\\\\ \\text{MAR2} & -0.789 & -1.59 & & & -0.232 & -0.44\\\\ \\hline \\end{array}<br \/>\n\\end{matrix} <\/p>\n<h2 style=\"text-align: center;\"><a id=\"displayTextt3\" href=\"javascript:togglecode('toggleTextt3','displayTextt3');\"><i><strong>R Code for Table 4.3<\/strong><\/i><\/a> <\/h2>\n<div id=\"toggleTextt3\" style=\"display: none\">\n<pre><strong>R-Code<\/strong>\r\nsummary(lm(LNFACE ~ LNINCOME+EDUCATION+NUMHH +MAR0+MAR2, data=Term2))\r\n<\/pre>\n<pre><strong>R-Code Output<\/strong>\r\nCall:\r\nlm(formula = LNFACE ~ LNINCOME + EDUCATION + NUMHH + MAR0 + MAR2, \r\n    data = Term2)\r\n\r\nResiduals:\r\n    Min      1Q  Median      3Q     Max \r\n-5.8875 -0.8505  0.1124  0.8468  4.5173 \r\n\r\nCoefficients:\r\n            Estimate Std. Error t value Pr(>|t|)    \r\n(Intercept)  3.39477    0.90019   3.771 0.000200 ***\r\nLNINCOME     0.45151    0.07872   5.736 2.61e-08 ***\r\nEDUCATION    0.20467    0.03862   5.299 2.42e-07 ***\r\nNUMHH        0.24770    0.06940   3.569 0.000424 ***\r\nMAR0        -0.55707    0.25929  -2.148 0.032574 *  \r\nMAR2        -0.78941    0.49532  -1.594 0.112169    \r\n---\r\nSignif. codes:  0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1\r\n\r\nResidual standard error: 1.513 on 269 degrees of freedom\r\nMultiple R-squared:  0.358,\tAdjusted R-squared:  0.3461 \r\nF-statistic:    30 on 5 and 269 DF,  p-value: < 2.2e-16\r\n<\/pre>\n<pre><strong>R-Code<\/strong>\r\nsummary(lm(LNFACE ~ LNINCOME+EDUCATION+NUMHH +MAR1+MAR2, data=Term2))\r\n<\/pre>\n<pre><strong>R-Code Output<\/strong>\r\nCall:\r\nlm(formula = LNFACE ~ LNINCOME + EDUCATION + NUMHH + MAR1 + MAR2, \r\n    data = Term2)\r\n\r\nResiduals:\r\n    Min      1Q  Median      3Q     Max \r\n-5.8875 -0.8505  0.1124  0.8468  4.5173 \r\n\r\nCoefficients:\r\n            Estimate Std. Error t value Pr(>|t|)    \r\n(Intercept)  2.83770    0.84882   3.343 0.000946 ***\r\nLNINCOME     0.45151    0.07872   5.736 2.61e-08 ***\r\nEDUCATION    0.20467    0.03862   5.299 2.42e-07 ***\r\nNUMHH        0.24770    0.06940   3.569 0.000424 ***\r\nMAR1         0.55707    0.25929   2.148 0.032574 *  \r\nMAR2        -0.23234    0.53283  -0.436 0.663155    \r\n---\r\nSignif. codes:  0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1\r\n\r\nResidual standard error: 1.513 on 269 degrees of freedom\r\nMultiple R-squared:  0.358,\tAdjusted R-squared:  0.3461 \r\nF-statistic:    30 on 5 and 269 DF,  p-value: < 2.2e-16\r\n<\/pre>\n<\/div>\n<div class=\"scbb-content-box scbb-content-box-gray\">[WpProQuiz 20]<\/div>\n<p> <div class=\"alignleft\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/chapter-4-multiple-linear-regression-ii\/4-1-the-role-of-binary-variables\/\" title=\"4.1 The Role of Binary Variables\">&#9668 Previous page<\/a><\/div><div class=\"alignright\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/chapter-4-multiple-linear-regression-ii\/4-2-statistical-inference-for-several-coefficients\/\" title=\"Protected: 4.2 Statistical Inference for Several Coefficients\">Next page &#9658<\/a><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We now return to the marital status of respondents from the Survey of Consumer Finances (SCF). Recall that marital status is not measured continuously but rather takes on values that falls into distinct groups that &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":3443,"menu_order":1,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-1i1","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/4961"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=4961"}],"version-history":[{"count":8,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/4961\/revisions"}],"predecessor-version":[{"id":4984,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/4961\/revisions\/4984"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3443"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=4961"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}