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Forced Expiratory Volume Regression Model

Katie Ruben

February 29, 2016 Forced expiratory volume (FEV1) is the volume of air forcibly expired in the first second of maximal expiration after a maximal inspiration. FEV1 is a significant parameter for identifying restrictive and obstructive respiratory diseases like asthma. It is a useful measure of how quickly full lungs can be emptied. A common clinical technique to measure this quantity is through Spirometry. Spirometry measures the rate at which the lung changes volume. The volume of the capacity of a lung is measured in liters. Results received from a spirometry test are dependent on the effort and cooperation between patients and examiners. These results depend heavily on the technicality of implementation as well as personal attributes of the patient [2]. Personal attributes that will help determine an accurate FEV1 score will be the patient's age, height, sex, and indication of being a smoker or non-smoker. The data used during this simulation comes from the Journal of Statistics Education Archive [3]. The data set consists of 4 variables, some of which are directly measured and some that are qualitative in nature. The data set is composed of a sample population consisting of 654 youth, male and female, aged between 3 and 19 years old from the East Boston area in the late 1970’s [5]. This data set contains 4 variables of measurement of children including age (years), height (inches), sex (male/female), and their self indication about being a smoker (yes/no). An investigation of the relationship between a child’s FEV1 and their current smoking status will be sought. It is important to note that the younger the child, the lower their FEV1 lung capacity will be due to the stature of their body alone. Therefore, in a normal case, the older the child is, the higher the lung capacity as their body grows. Another good measure of lung capacity is looking at the ratio between FEV1 and FVC. Forced vital capacity (FVC) is the maximum total volume of air expired after a maximal deep breath in, which takes 6 seconds to fully expire. A normal ratio value for a person without pulmonary obstruction is between 80% and 120% [1]. A percentage lower than 80% is indicative of obstructive lung functions. Since a predicted FVC value is not provided in the data set, one could use known formulas to calculate this value for male and female children based off of their personal attributes [4]. However, in using the data provided to calculate the predicted FVC value will result in using the parameters for each child twice. Once in the predicted FVC formula and once again when I perform a regression analysis. This would not be a good idea. Therefore, I will exclude the the FEV1 to FVC ratio from my analysis, but it is good background information in interpreting a person’s lung function. In order to analyze this data, I will use our predictor variables to construct a linear regression model for predicting FEV1 values. Upon initial fittings, I will analyze the model and look for any initial predicting issues. Additionally, I will interpret the analysis of the data in order to

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describe the meaning of FEV1 to each of the predictor variables dependent on the multiple regression model chosen. I can determine the correlation between each of the predictor variables and my regression fittings by looking at plots of each predictor variable and the fitted FEV1 calculated. Additionally, I can compute the correlation matrix to begin an initial evaluation of strongly correlated variables in the data. I will look into determining if there are any multicollinearity problems in our data. After determining the most necessary and possibly unnecessary variables, I will try to find an appropriate regression model for the predictor variables provided in the data set. Further, I will test for any possible outliers in the data that would be significantly influencing the regression. In the case that we find significant outliers, I will remove this data and try to create a new regression. In this analysis of data, I will use multiple regression techniques to try to find the best fit for our given data set. We will test for multicollinearity problems, the significance of our regression coefficients, and assess for possible outliers.

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1   Background

Forced expiratory volume (FEV1) is the volume of air forcibly expired in the first second of maximal expiration after a maximal inspiration. FEV1 is a significant parameter for identifying restrictive and obstructive respiratory diseases like asthma. It is a useful measure of how quickly, full lungs can be emptied. A common clinical technique to measure this quantity is through Spirometry. Spirometry measures the rate at which the lung changes volume. The volume of the capacity of a lung is measured in liters. Results received from a spirometry test are dependent on the effort and cooperation between patients and examiners. These results depend heavily on the technicality of implementation as well as personal attributes of the patient [2]. Personal attributes that will help determine an accurate FEV1 score will be the patient's age, height, sex, and indication of being a smoker or non-smoker.

VARIABLE DESCRIPTION

𝒀 FEV1 (liters) 𝑿𝟏 Age (years) 𝑿𝟐 Height (inches) 𝑿𝟑 Sex (male or female) 𝑿𝟒 Smoker (yes or no)

Table 1: Variable Descriptions

For our model prediction analysis, we use a data set containing four variables. This data set is from The Journal of Statistical Education and publically shared by Michael Kahn [3] with the approval of Bernard Rosner who published the data in 1999 in Fundamentals of Biostatistics [5]. The data set is composed of a sample population consisting of 654 youth, male and female, aged between 3 and 19 years old from the East Boston area in the late 1970’s. An investigation of the relationship between a child’s FEV1 and their current smoking status will be sought as well as any other comparisons between predictor variables. The variable descriptions can be found in table 1. The indication of smoking for predictor variable 𝑋4, is qualitative data about each child. The child made an indication if they, themselves were smokers or not while the data was being collected. It is important to note that the younger the child, the lower their FEV1 lung capacity will be due to the stature of their body alone. Therefore, in a normal case, the older the child is, the higher the lung capacity as their body grows. As seen in Figure 1, the taller the child is then the higher their FEV1. In addition, Figure 1 shows that in general as the child gets older their FEV1 increase however further investigation is needed into the interpretation of the FEV1 versus age scatterplot. There may exist other factors that result in a drop of FEV1 as the children reach puberty. We aim to find our best linear regression model for predicting FEV1 based off of our four predictor variables; age, height, sex, and indication of smoking. In our model building process we will want to determine if we can predict FEV1 using less measurements.

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Figure 1: Scatterplots of height and age versus FEV1.

In this paper, we begin by using our training data for model building in Section 2. We will begin with a preliminary model and use different techniques to determine other possible models. These models will then be tested to determine what our final prediction model should be. We then use our data to determine if our final model can be validated in Section 3. In Section 4, we end with a discussion of our findings and possible future analyses.

2   Model Building 2.1.1 Preliminary Model

To begin our model building process, we start by creating our preliminary model for our data set. The preliminary equation that we use is:

𝑌* = .067635𝑋* + .102853𝑋6 + .189609𝑋8 − .113826𝑋: − 4.396799

Figure 2: Residual Analysis for Normality.

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Each of the predictor variables in our model were significant at the 𝛼 = .01 level except for X4. However, I choose to leave this variable in the data as X4 represents smoking. The p-value of each variable was low which indicates significance. (R output can be found in Appendix A.1).

By analyzing our residuals, we can determine the normality and homoscedacity for our model. By plotting the normal probability plot for our residuals, seen in Figure 2, we see that we may have normality issues. The normality plot shows heavy tails on both ends. To ensure normality, we perform the Shapiro-Wilk and Kolmogorov-Smirnov tests for normality (see Appendix A.1). At the 𝛼 = .05 level, we conclude from both tests that the data is not from the normal distribution because we must reject null hypothesis if the p-value is larger than .05. The p-value in the Shaprio-Wilk Test is small, however, the p-value in the Kolmogorov-Smirnov test is greater than .05. Hence, the tests don’t agree and normality is rejected. In addition to normality, we test the residuals for constant variance. We plot the fitted values versus the residuals; these plots can be found in Figure 3. Based on this residual plot, we can conclude that we do not have constant error variance due to the megaphone type distribution of our data. The Breusch-Pagan test and Brown-Forsythe test both also confirms that that we do not have constant error variance (R output can be found in Appendix A.1).

Figure 3: Residual Analysis for Homoscedacity in model 1; 𝑌*.

2.1.2   Transformed Preliminary Models

2.1.2.1 Model 2

Due to lack of constant error variance and normality, we test our data to determine an appropriate transformation on the response variable. Since unequal variance and non-normality of error terms frequently appear together we can remedy this by performing a transformation on 𝑌. The transformation is 𝑌< = 𝑙𝑜𝑔*@(𝑌), which results in our second model.

𝑌6 = .009811𝑋* + .018629𝑋6 + .017802𝑋8 − .026152𝑋: − .843013

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By plotting the normal probability plot for our residuals, seen in Figure 4, we see that we may have normality issues once again. We can apply the S-W Test and K-S Test to check for normality, but the same issue as in model 1 occurs again (R output can be found in Appendix A.2). However, model 2 now shows that there is constant error variance based on the residual plot of 𝑌6, seen in Figure 4. The B-P test also concludes constant error variance in 𝑌6 for the p-value is large. Hence, we accept the null hypothesis that the second model has constant error variance.

. Figure 4: Residual analysis for normality and constant error variance in Model 2.

2.1.2.2   Model 3

Due to the lack of normality, we test our data to determine if there is an appropriate transform. In order to do this we use the Box-Cox method on our data of model 1. We conclude from the Box-Cox method, that a transformation for the lambda value should be 𝜆 = .1 for our preliminary model 𝑌*, as seen in Figure 5. The transformed model can be found in Appendix A.2. We then test the new transformed model for normality and homoscedacity. We come to the same conclusion as in model 2. The transformation is still non-normal but, does have constant error variance.

𝑌8 = .0025108𝑋* + .0046634𝑋6 + .0048190𝑋8 − .0064011𝑋: + .7851634

Figure 5: Box Cox output for 𝜆 is determined to be . 1

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2.1.3 Testing for Multicollinearity Due to the nature of the variables included in this study, we assume there will be multicollinearity. We expect to see multicollinearity between age, height, and sex. In the correlation matrix of the predictor variables and response variable we expect to see high values. The correlation matrix in Appendix A.3 indicates a strong correlation between the indicated predictor variables stated earlier. We can see these strong correlations and underlying linear relationships in the correlation plot in Appendix A.3. Additionally, we calculate the variance inflation factors (VIF). The preliminary model that we choose to continue working with is model 2 where we performed a log transformation on our response variable y. The variance inflation factors do not indicate strong multicollinearity in the preliminary model 2 since no variables show large (>10) VIF values. Although, VIF doesn’t suggest multicollinearity, we will still use different methods for model selection to choose the best models for calculating FEV1, our response variable.

2.1.3   Testing for Outliers

Before we move into the model selection process, we will run some preliminary tests to see if our data set contains outliers that need to be taken care of. By looking at Figure 6, we can determine several noticeable points that may be considered outliers in our data.

Figure 6: Analysis for Outliers of 𝑌6.

In R, we run the influence measures command which signifies 34 data points that are potential significant influencers. By looking at the residuals vs. leverage graph in Figure 6, we see that

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point 323, 44, and 224 are labeled values. Note that the scale on the leverage axis is extremely small. Hence, in reality none of these data points are truly that far apart in order to skew our regression model. I will test these three values using the influence measures of DEFITS, Cook’s Distance, H-matrix, and DFBETA.

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323 -0.53721 5.62e-02 0.02865 -3.94e-01 -1.17e-02 3.16e-01 -2.51e-01 -1.05e-01

224 -0.51478 5.12e-02 0.02091 0.02865 -3.46e-03 -2.71e-01 2.75e-01 2.10e-01 44 -0.55968 6.12e-02 0.03405 1.21e-01 1.14e-01 -1.39e-01 1.52e-01 -4.51e-01

Table 1: Potential Outliers

If the DEFITS value is greater than 1, we conclude that the point is influential. Above, these three points are not influential according to this criterion. In order to assess influential points based off of the COOKS distance, I will need to find the F-distribution for 𝐹 5,327 − 5 = 𝐹(5,322) for each of the COOK values. If the percentile value is less than 10 or 20 percent, then the case has little apparent influence on the fitted values. When n=323, we get that .0562 is the .2th percentile of the distribution. When n=224, we get that .0512 is the .16th percentile of the distribution. When n=44, we get that .0612 is the .25th percentile of the distribution. Since all of these percentile values are less than 10 or 20 percent, we conclude again that these cases are non-influential. In order to assess if outliers exists in relation to the ℎGG, I will look to see if ℎGG >

6IJ

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86M= .030581. Based on the cases of interest in the chart above, we see that case 44 would

be considered an outlier. However, the other influence measures do not suggest this point as an outlier. In order to assess influential points based off of DFBETAS we look to see if the absolute value of number presented for each case exceeds 1. If it exceeds 1 then that case might be an influential point. Again, none of these points leads to an outlier. The code used in R to get the output for this list of influential measures is located in Appendix A.1. The additional, potential influential points can be assessed by looking at the code. 2.2   Model Selection

Now we will use multiple methods to determine possible subsets of predictor variables to use in a new model based off of preliminary model 2. We will discuss our selection methods and new potential models below.

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Using R’s leaps package, we can find appropriate subsets to use for model prediction. Using the scale of CP Mallow values, shown in Appendix A.4, we find the same best possible subset of variables as presented in 𝑌6. Thus, we look for other model selections. We use the same method but, use adjusted R-squared scale to determine the subset collection now (shown in Appendix A.4). This method yields the subsets {𝑋*, 𝑋6}, {𝑋*, 𝑋6, 𝑋8, }, and {𝑋*, 𝑋6, 𝑋:}. Note, that all three of these subsets have the same adjusted R-squared value. I want to find multiple subsets of prediction models to compare which model is better. Using these two new subsets, we get the following three new models:

𝑌6,* =. 0079265𝑋*+.0192128𝑋6 − .8537001

𝑌6,6 =. 0090268𝑋* + .0192252𝑋6 − .0293616𝑋: − .8623788

𝑌6,8 =. 0089099𝑋* + .0185659𝑋6 + .0193566𝑋8 − .8336753 In addition, we used the forward and backward AIC stepwise method in R. The model that we come up with in R has the same subset of variables as presented in 𝑌6. This can be seen in the R output in Appendix A.4. Since, our data set only contains four variables, this is not uncommon for the predicted model to be the same as our preliminary model if all of the variables are important to the calculation of the response. With each method, we have varying subsets of predictor variables. In order to determine the best model, we compute 𝑅QRS6 , 𝑉𝐼𝐹, 𝐴𝐼𝐶, 𝐵𝐼𝐶, and PRESS for each model. We aim to find a model with the smallest 𝑉𝐼𝐹, 𝐴𝐼𝐶, 𝐵𝐼𝐶, and PRESS as well as the largest 𝑅QRS6 ; these values are highlighted in the Table 2.

Model 𝑅QRS6  𝑉𝐼𝐹 𝐴𝐼𝐶 𝐵𝐼𝐶 𝑃𝑅𝐸𝑆𝑆 𝑌6 .813 1.941479 -1813.209 -1794.259 1.280946 𝑌6,* .8072 2.561182 -1805.266 -1793.896 1.309714 𝑌6,6 .8099 2.150455 -1808.889 -1793.73 1.297981 𝑌6,8 .811 2.144651 -1810.701 -1795.541 1.288214

Table 2: Four potential FEV1 Models

As seen in the Table 2, each model yields around the same 𝑅QRS6 . From the mean VIF values, we see that there is no serious multicollinearity problem since no values are greater than 10. Additionally, the AIC and BIC values for each model do not vary significantly. From the table, we see that 𝑌6 appears to be the best in four of the five tests performed in Table 2. Thus, we will use this as our final FEV1 model. This model is the transformed preliminary model we had initial constructed. We have just verified that this model has the greatest potential for predicting accurately FEV1 values. We will validate our final model in Section 3. Make note that since no predictor variables are being dropped, we will not need to conduct a partial F test. However, if we were to drop a variable for our final model selection, we would want to ensure that it is sufficient to do so. In order to ensure it is sufficient, we would run a generalized F test on our full

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model with our reduced model. An example of this has been provided in Appendix A.2 section 2.2.1. Thus, from our model selection process, we finalize that our final model will be:

𝑌6 = .009811𝑋* + .018629𝑋6 + .017802𝑋8 − .026152𝑋: − .843013 The final model was formed using our training data; we proceed with the validation of this model using the validation data in the Section 3. 3 Model Validation In order to determine the prediction ability of our final model, we use our validation data. We use the remaining data from our set to validate the final model to predict FEV1. For the remaining data, we run the linear regression for the model using all of the predictor variables, {𝑋*, 𝑋6, 𝑋8, 𝑋:}. Our regression yields the summary output found in Figure 7 for the validation data.

Figure 7: Summary of Final Model regressed with validation data (right). Summary of Final Model regressed with model data (left).

When re-estimating the model with the validation data we have 𝑀𝑆𝑃𝑅 = .004169. By comparing this value to the MSE based on the model-building data we see that the values are fairly close. In the model-building data regression model, 𝑀𝑆𝐸 = .003847. This is a good indication that the selected regression model is not seriously biased and gives an appropriate indication of the the predictive ability of the model. Any R outputs corresponding to Section 3, model validation, are located in Appendix A.5. The results of both summaries in Figure 7, are consistent and hence our model seems to validate our validation data with our model data. The coefficients for the corresponding predictor variables are very similar. This would lead us to believe that our model is accurately predicting FEV1. Additionally, for our validation model, we see that we do have a strong correlation; our adjusted R-squared is about 81% for both sets of data regressed with the chosen model. Thus, our model is a good fit for our data.

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Figure 8: Final Regression Model on Full Data Set

Now that the data has been validated, we have used the entire data set to estimate the final regression model. As you can see in Figure 8, the coefficients of each predictor variable are close to the coefficients for the regression performed on the model and validation data sets. Our final model is:

𝑌6 = .0101569𝑋* + .0185860𝑋6 + .0127332𝑋8 − .0200069𝑋: − .8442677 4 Discussion The goal of this analysis was to predict FEV1 values in children of varying age, height, sex, and indication of being a smoker. We started with 327 children in our data that was set aside strictly for modeling the regression. We wanted to find a model that would predict FEV1 based off of this subset of participants from our whole dataset. Keep in mind, the whole data set contained 654 participants. We began our analysis with training data and created a preliminary model that included four variables. We tested our residuals for normality and homogeneity. Next, we determined that we did not have normality or constant error variance. In order to attempt to fix this problem, we performed several different transformations on our response variable. In both of the transformations performed, we found that we would get constant error variance but, still have non-normality. We decided to continue with the model that took the log transformation on the response variable. We had expected that there would be multicollinearity issues with our data set by strictly looking at the correlation scatterplot provided in Appendix A.3. However, when we calculated the VIF value, no such issues arose. We also tested our preliminary model for outliers by looking at a variety of influence measures. We were able to conclude that no strong indication of an outlier existed from these influence measures. From our model selection process, we found a model that would estimate FEV1. Our final model actually consisted of all of the four predictor variables. Again, this is not unusual since each of these predictor variables were significant. Using our final model, that was produced from the modeling data set, we used our validation data to determine the validity of our model. We found

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that our final model had a high adjusted R-squared, indicating that our model is an appropriate fit. Further indication that our model was appropriate came from the comparison of MSPR and MSE as discussed in Section 3 of this paper. In order to further improve this model, we should try to account for the normality issues. If we can appropriately transform the data, we could potentially create a model with a better fit. Although our initial fit is good, there is room for improvement. We are interested in finding if there are interaction variables that could produce a better fit for our regression model. An example of such an interaction variable could consist of looking at X1 times X4 or any other variation. We could also look at transforming our predictor variables by taking the square, cubic, etc. of each and do some comparisons of models. In the end, the analysis of this data has shown us that the FEV1 value for the children in this data set is dependent on all four predictor variables. In addition, we can draw the conclusion that FEV1 is affected by age, height, sex, and smoking. Individual correlations between FEV1 and the predictor variables can be sought; however, the purpose of this analysis was to use multiple linear regression.

Reflection on Project: If I were to do another project like this, I would have chosen a data set with more than four predictor variables. It would have been more interesting to see which variables I would add or drop if I had 10 or more predictor variables. However, I have learned the process of model selection from this small subset of predictor variables in the data set I have chosen.

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Appendix A Reference for Model Building

A.1 Preliminary Model Model One: Y1~x1+x2+x3+x4: s1 Call: lm(formula = Y1 ~ X1 + X2 + X3 + X4) Residuals: Min 1Q Median 3Q Max -1.31452 -0.22975 0.00576 0.24448 1.49585 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -4.396799 0.310140 -14.177 < 2e-16 *** X1 0.067625 0.012641 5.350 1.68e-07 *** X2 0.102853 0.006546 15.713 < 2e-16 *** X3 0.189609 0.046818 4.050 6.43e-05 *** X4 -0.113826 0.081545 -1.396 0.164 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.4089 on 322 degrees of freedom Multiple R-squared: 0.7784, Adjusted R-squared: 0.7756 F-statistic: 282.7 on 4 and 322 DF, p-value: < 2.2e-16

Tests  for  Normality:  𝐻@:Residuals are normally distributed. 𝐻Q:Residuals are not normally distributed. Significance level ∝=. 𝟎𝟓 >ks.test(residuals(m1),"pnorm", mean=0, sd=sd(residuals(m1))) KS-Test: One-sample Kolmogorov-Smirnov test data: residuals(m1) D = 0.054421, p-value = 0.2875 >.05 alternative hypothesis: two-sided >shapiro.test(residuals(m1)) Shapiro Test: Shapiro-Wilk normality test data: residuals(m1) W = 0.9889, p-value = 0.01356 <.05

Tests  for  Constant  Error  Variance:  𝐻@:Residuals have constant variance. (accept if 𝑝 − 𝑣𝑎𝑙𝑢𝑒 >. 𝟎𝟓 𝐻Q:Residuals do not have constant variance. Significance level ∝=. 𝟎𝟓 Brown Forsythe Test: In order to split my data into two groups, I looked at the age of my participants. Group one contains 155 observations for their Age<=9 and group two contains 172 observations for their Age>=10. library(car) data.BF1<- modeldata[order(modeldata[,1]),] X1.newBF1<-data.BF1[,1] X2.newBF1<-data.BF1[,3] X3.newBF1<-data.BF1[,4] X4.newBF1<-data.BF1[,5] Y.newBF1<-data.BF1[,2] z.BF1<-residuals(lm(Y.newBF1~X1.newBF1+X2.newBF1+X3.newBF1+X4.newBF1)) g1<-rep(0,155) g2<-rep(1,172) group<-as.factor(c(g1,g2)) leveneTest(z.BF1,group) Levene's Test for Homogeneity of Variance (center = median) Df F value Pr(>F) group 1 20.906 6.867e-06 *** 325 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 BP Test: library(lmtest) bptest(Y1~X1+X2+X3+X4,studentize = FALSE) Breusch-Pagan test data: Y1 ~ X1 + X2 + X3 + X4 BP = 48.145, df = 4, p-value = 8.803e-10

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A.2 Transformed Model

2.2.1 Model 2

Generalized F-Test Example:

The F-value is large so this suggests that we would not want want to drop the predictor variable X4.

Tests  for  Normality:  𝐻@:Residuals are normally distributed. 𝐻Q:Residuals are not normally distributed. Significance level ∝=. 𝟎𝟓

Tests  for  Constant  Error  Variance:  𝐻@:Residuals have constant variance. (accept if 𝑝 − 𝑣𝑎𝑙𝑢𝑒 >. 𝟎𝟓 𝐻Q:Residuals do not have constant variance. Significance level ∝=. 𝟎𝟓

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Residuals

Tests  for  Normality:  𝐻@:Residuals are normally distributed. 𝐻Q:Residuals are not normally distributed. Significance level ∝=. 𝟎𝟓

Tests  for  Constant  Error  Variance:  𝐻@:Residuals have constant variance. (accept if 𝑝 − 𝑣𝑎𝑙𝑢𝑒 >. 𝟎𝟓 𝐻Q:Residuals do not have constant variance. Significance level ∝=. 𝟎𝟓

16

A.3 Multicollinearity

V1-Age, V2-FEV, V3-Height, V4-Sex, V5-Smoker

> vif(lm(Y.new1~X1+X2+X3+X4)) X1 X2 X3 X4

2.797353 2.717221 1.071498 1.179844

17

A.4 Model Selection

Forward AIC Backward AIC

Cp

(Intercept) X1

X2

X3

X4

1400

1300

1300

410

400

360

350

30

30

29

28

13

9.3

7.5

5

adjr2

(Intercept) X1

X2

X3

X4

0.027

0.054

0.089

0.58

0.58

0.61

0.61

0.8

0.8

0.8

0.8

0.81

0.81

0.81

0.81

Calculations to Compare Models:

18

A.5 Model Validation Validation Model

Final Model

19

References [1] Barreiro, T. J., D.O., & Perillo, I., M.D. (2004, March). An Approach to Interpreting Spirometry. Retrieved February 10, 2016, from http://www.aafp.org/afp/2004/0301/p1107.html [2] Kavitha, A., Sujatha, C. M., & Ramakrishnan, S. (2010). Prediction of Spirometric Forced Expiratory Volume (FEV1) Data Using Support Vector Regression. Measurement Science Review, 10(2). Retrieved February 8, 2016, from http://www.measurement.sk/2010/S1/Kavitha.pdf [3] Michael, K. (2005). An Exhalent Problem for Teaching Statistics. Retrieved February 10, 2016, from http://www.amstat.org/publications/jse/v13n2/datasets.kahn.html Journal of Statistics Education Volume 13, Number 2 (2005), www.amstat.org/publications/jse/v13n2/datasets.kahn.html [4] Knudson, RJ, et. al. The Maximal Expiratory Flow-Volume Curve Normal Standards, Variability, and Effects of Age. Am. Rev. Respir. Dis. 113:589-590, 1976. Retrieved February 8, 2016, from http://cysticfibrosis.com/forums/topic/fev1/ [5] Rosner, B. (1999), Fundamentals of Biostatistics, 5th ed., Pacific Grove, CA: Duxbury.