SMHS LinearModeling LMM Assumptions - Revision history

Dinov: /* Hands-on Activity */

2017-10-07T21:05:09Z

‎Hands-on Activity

Imoubara: /* Example of Logistic Regression */

2016-05-23T13:58:56Z

‎Example of Logistic Regression

Imoubara: /* Types of Data Analyses16 */

2016-05-23T13:58:12Z

‎Types of Data Analyses16

Dinov at 17:28, 21 May 2016

2016-05-21T17:28:28Z

Dinov: /* Linear mixed effects analyses - Mixed Effect Model Assumptions */

2016-05-21T17:26:06Z

‎Linear mixed effects analyses - Mixed Effect Model Assumptions

Pineaumi: /* Linear mixed effects analyses - Mixed Effect Model Assumptions */

2016-05-19T17:12:02Z

‎Linear mixed effects analyses - Mixed Effect Model Assumptions

Pineaumi: /* Mixed Effects Logistic Regression */

2016-05-19T14:56:58Z

‎Mixed Effects Logistic Regression

Pineaumi: /* Mixed Effects Logistic Regression */

2016-05-19T14:46:44Z

‎Mixed Effects Logistic Regression

Show changes

Pineaumi: /* Types of Data Analyses16 */

2016-05-19T13:51:09Z

‎Types of Data Analyses16

Pineaumi: /* Types of Data Analyses16 */

2016-05-19T13:48:32Z

‎Types of Data Analyses16

@@ Line 29: / Line 29: @@
 ===Hands-on Activity===
-Use these cancer data (http://www.ats.ucla.edu/stat/data/hdp.csv), representing cancer phenotypes and predictors (e.g., "IL6", "CRP", "LengthofStay", "Experience") and outcome measures (e.g., remission) collected on patients, nested within doctors (DID) and within hospitals (HID). To fit a mixed model (http://www.ats.ucla.edu/stat/r/dae/melogit.htm) and examine remissions as cancer outcomes.
+Use these cancer data (https://stats.idre.ucla.edu/stat/data/hdp.csv), representing cancer phenotypes and predictors (e.g., "IL6", "CRP", "LengthofStay", "Experience") and outcome measures (e.g., remission) collected on patients, nested within doctors (DID) and within hospitals (HID). To fit a mixed model (http://dev2.hpc.ucla.edu/stat/stata/dae/melogit.htm) and examine remissions as cancer outcomes.
 This lung cancer dataset includes a variety of outcomes collected on patients, nested within doctors, who are in turn nested within hospitals. Doctor level variables include experience.
-  hdp <- read.csv("http://www.ats.ucla.edu/stat/data/hdp.csv")
+  hdp <- read.csv("https://stats.idre.ucla.edu/stat/data/hdp.csv")
   hdp <- within(hdp, {
   Married <- factor(Married, levels = 0:1, labels = c("no", "yes"))

@@ Line 284: / Line 284: @@
 ''Intercept'' = -4.1030 and  ''SE'' = 0.7583.
-<li>Probability of surviving heart surgery CO =1/(1+exp(-(-4.1030+0.7583×SE)))</li>
+*Probability of surviving heart surgery CO =1/(1+exp(-(-4.1030+0.7583×SE)))
-<li>For example, for a patient who is operated by a surgeon with 200 hours of operating experience (SE=2), we plug in the value 2 in the equation to get an estimated probability of survival, <u><b>p=0.07:</b></u></li>
+*For example, for a patient who is operated by a surgeon with 200 hours of operating experience (SE=2), we plug in the value 2 in the equation to get an estimated probability of survival, <u><b>p=0.07:</b></u>
 <b>SE=2;</b> <BR>
@@ Line 292: / Line 292: @@
 [1] <b><u>0.07001884</u></b>
-<li>Similarly, a patient undergoing heart surgery with a doctor that has 400 operating hours experience (SE=4), the estimated probability of survival is <b><u>p=0.26:</u></b></li>
+*Similarly, a patient undergoing heart surgery with a doctor that has 400 operating hours experience (SE=4), the estimated probability of survival is <b><u>p=0.26:</u></b>
 <b>SE=4;</b> CO =1/(1+exp(-(-4.1030+0.7583*SE))); CO <BR>
 <font color="blue">> CO<BR>

@@ Line 76: / Line 76: @@
 ===Types of Data Analyses<sup>16</sup>===
-<li>Mixed effects logistic regression, the focus of this page. </li>
+*Mixed effects logistic regression, the focus of this page.
-<li>Mixed effects probit regression is very similar to mixed effects logistic regression, but it uses the normal CDF instead of the logistic CDF. Both model binary outcomes and can include fixed and random effects. (Note: This link function, aka Probit link, defined in the 1930’s by biologists studying the dosage-cure rate link, refers to the “probability unit”. It’s kind of the inverse CDF, of the model: if Y = Φ(Xβ+ ε), then Probit link = Φ^(-1) (Y). </li>
+*Mixed effects probit regression is very similar to mixed effects logistic regression, but it uses the normal CDF instead of the logistic CDF. Both model binary outcomes and can include fixed and random effects. (Note: This link function, aka Probit link, defined in the 1930’s by biologists studying the dosage-cure rate link, refers to the “probability unit”. It’s kind of the inverse CDF, of the model: if Y = Φ(Xβ+ ε), then Probit link = Φ^(-1) (Y).
-<li>Fixed effects logistic regression is limited in this case because it may ignore necessary random effects and/or non-independence in the data. </li>
+*Fixed effects logistic regression is limited in this case because it may ignore necessary random effects and/or non-independence in the data.
-<li>Fixed effects probit regression is limited in this case because it may ignore necessary random effects and/or non-independence in the data.</li>
+*Fixed effects probit regression is limited in this case because it may ignore necessary random effects and/or non-independence in the data.
-<li>Logistic regression with clustered standard errors. These can adjust for non-independence but does not allow for random effects.</li>
+*Logistic regression with clustered standard errors. These can adjust for non-independence but does not allow for random effects.
-<li>Probit regression with clustered standard errors. These can adjust for non-independence but does not allow for random effects.</li>
+*Probit regression with clustered standard errors. These can adjust for non-independence but does not allow for random effects.
 ===Mixed Effects Logistic Regression===

@@ Line 28: / Line 28: @@
 Often fixed effects are expected to exhaust the population of interest, or the levels of a factor. In the MLB study the factor “Team” may not exhaust the space as there are other teams/leagues. However, for MLB at a fixed time, the “Team” factor may be fully exhaustive. Same with Height. Random effects represent sub-samples from the population of interest and may not “exhaust” the population as more players or teams could be included in the study. The levels of random factor may only represent a small sub-subset of all levels of the factor.
-<b>Hands-on Activity</b>
+===Hands-on Activity===
 Use these cancer data (http://www.ats.ucla.edu/stat/data/hdp.csv), representing cancer phenotypes and predictors (e.g., "IL6", "CRP", "LengthofStay", "Experience") and outcome measures (e.g., remission) collected on patients, nested within doctors (DID) and within hospitals (HID). To fit a mixed model (http://www.ats.ucla.edu/stat/r/dae/melogit.htm) and examine remissions as cancer outcomes.
@@ Line 215: / Line 214: @@
   VarCorr(m1); anova(m1); intervals(m1)
-<B>Example of Logistic Regression: </B>
+===Example of Logistic Regression===
 (1)	Logistic curve:
 $y=f(x)=\frac{1}{1+e^{-x}}$,
@@ Line 249: / Line 246: @@
 |}
 </center>
   mydata <- read.csv('https://umich.instructure.com/files/405273/download?download_frd=1&verifier=AOny2eq3wF7WqWAV5YsT6e9zakRTEbZpcuNYRdtM')  # 01_HeartSurgerySurvivalData.csv
@@ Line 325: / Line 321: @@
 {| class="wikitable" style="text-align:center; width:35%" border="1"
 |-
 | ||Estimate Std.||Error||z value||Pr (>z) Wald
@@ Line 385: / Line 380: @@
 The distribution of the LRT is chi-squared with degrees of freedom equal to the number of parameters that are reduced (i.e., the number of variables removed from the model). In our case,
-<mark>$LRT \sim X^2_{df=2}$ as we have an intercept and one predictor (SE), and the null model is empty (no parameters). FIX ~SIGN, PLEASE! </mark>
+$LRT \sim X^2_{df=2}$ as we have an intercept and one predictor (SE), and the null model is empty (no parameters).
 ===Footnotes===

@@ Line 385: / Line 385: @@
 The distribution of the LRT is chi-squared with degrees of freedom equal to the number of parameters that are reduced (i.e., the number of variables removed from the model). In our case,
-<mark>$LRT\~X^2_{df=2}$ as we have an intercept and one predictor (SE), and the null model is empty (no parameters). FIX ~SIGN, PLEASE! </mark>
+<mark>$LRT \sim X^2_{df=2}$ as we have an intercept and one predictor (SE), and the null model is empty (no parameters). FIX ~SIGN, PLEASE! </mark>
+===Footnotes===
-<sup>15</sup> http://cran.r-project.org/web/packages/influence.ME/influence.ME.pdf<BR>
+* <sup>15</sup> http://cran.r-project.org/web/packages/influence.ME/influence.ME.pdf
-<sup>16</sup> http://www.ats.ucla.edu/stat/mult_pkg/glmm.htm
+* <sup>16</sup> http://www.ats.ucla.edu/stat/mult_pkg/glmm.htm
-<sup>17</sup> http://wiki.socr.umich.edu/index.php/SMHS_OR_RR
+* <sup>17</sup> http://wiki.socr.umich.edu/index.php/SMHS_OR_RR
 ==Next See==

@@ Line 48: / Line 48: @@
 <center>[[Image:SMHS_LinearModeling_Fig35.png|500px]]</center>
-Are there strong linear relations among the continuous variables? Examine CancerStage and LengthofStay closer. The area of bubbles are proportional to the number of observations with the corresponding values.
+Are there strong linear relations among the continuous variables? Examine CancerStage and LengthofStay closer. The area of bubbles is proportional to the number of observations with the corresponding values.
 Violin plots may be used for continuous predictors. We can render all raw data separated by CancerStage. To reduce overlaying, we can add some random noise (along the x axis) or alternatively set the alpha opacity level.
@@ Line 390: / Line 390: @@
 <sup>15</sup> http://cran.r-project.org/web/packages/influence.ME/influence.ME.pdf<BR>
 <sup>16</sup> http://www.ats.ucla.edu/stat/mult_pkg/glmm.htm
 ==Next See==

@@ Line 379: / Line 379: @@
 <b>Side-note</b>: The LRT compares the data fit of two models. For instance, removing predictor variables from a model will reduce model quality (i.e., a model will have a lower log likelihood). To statistically assess whether the observed difference in model fit is significant, the LRT compares the difference of the log likelihoods of the two models. When this difference is statistically significant, the full model (the one with more variables) is a better fit to the data, compared to the reduced model. LRT is computed from the log likelihoods of the models:
----------MISSING FORMULA------------
+$LRT=-2ln\Big(\frac{L(m1)}{L(m2)}\Big)=2(ll(m2)-ll(m1)) $,
-<i>m1</i> and <i>m2</i> are the reduced and the full models, respectively, <i>L(m1)</i> and <i>L(m2)</i> denote the likelihoods of the 2 models, and <i>ll(m1)</i> and <i>ll(m2)</i> represent the log likelihood (natural log of the model likelihood.
+Where <i>m1</i> and <i>m2</i> are the reduced and the full models, respectively, <i>L(m1)</i> and <i>L(m2)</i> denote the likelihoods of the 2 models, and <i>ll(m1)</i> and <i>ll(m2)</i> represent the log likelihood (natural log of the model likelihood.
 The distribution of the LRT is chi-squared with degrees of freedom equal to the number of parameters that are reduced (i.e., the number of variables removed from the model). In our case,
-----MISSING FORMULA----- as we have an intercept and one predictor (SE), and the null model is empty (no parameters).
+<mark>$LRT\~X^2_{df=2}$ as we have an intercept and one predictor (SE), and the null model is empty (no parameters). FIX ~SIGN, PLEASE! </mark>

← Older revision		Revision as of 13:48, 19 May 2016
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−	<sup>15</sup> http://cran.r-project.org/web/packages/influence.ME/influence.ME.pdf	+	<sup>15</sup> http://cran.r-project.org/web/packages/influence.ME/influence.ME.pdf<BR>
		+	<sup>16</sup> http://www.ats.ucla.edu/stat/mult_pkg/glmm.htm

	==Next See==		==Next See==