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Created: Apr 29, 2016
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Wk 15 Predicting Height--revision
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Introduction:

The height of an adult is dependent on several factors including gender, genetics, and overall health and nutrition.  It is estimated that 70% of height is determined by genetics while 30% is determined by other environnmental factors such as nutrition, exercise, and any underlying health conditions (www.kidsgrowth.com).

Methods:

This study was done to determine which measurements best predicted height (standing) in women and height (sitting) in women.  Measurements were taken for 33 women, in centimeters, of the upper arm length, forearm length, hand length, upper leg length, lower leg length and foot length.  A multiple linear regression in StatCrunch was used to analyze the data.  Following the MLR analysis, the data was used to predict height in two unknown subjects given measurements for upper arm, forearm, hand, upper leg, lower leg and foot lengths.

The purpose of our study was to assess whether or not there is a correlation between the various measurements and height (standing) and height (sitting).   Significance value was set at 0.05.

Analysis:

To assess normality, a QQ plot of residuals was performed for height (standing) and height (sitting).   For each QQ plot, the points are along the diagonal line, therefore normality is assumed.

Result 1: Wk 14 QQ Plot Residuals Height Standing   [Info]
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Result 2: Wk 14 QQ Plot Residuals vs. Height Sitting   [Info]
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To assess variability, a scatterplot of residuals vs. predicted values for height (standing) and height (sitting).  For each scatterplot, there is random scatter of the points with no particuar pattern, therefore we assume variability.

Result 3: Wk 14 Scatter Plot Residuals vs. Predicted (Height Standing)   [Info]
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Result 4: Wk 14 Scatter Plot Residuals vs. Predicted (Height Sitting) 2   [Info]
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With normality and variability assumed for both height (standing) and height (sitting), a correlation matrix was performed.  From the correlation matrix, for height (standing), all measurements (upper arm length, forearm length, hand length, upper leg length, lower leg length, and foot length) were siginificant, with p-values <0.05, to give evidence of a correlation between those measurements and height (standing).  However, with significant p-values among many of the variables, there is evidence of confounding.  For height (sitting), only foot length, with a significant p-value of 0.04, provides evidence of a correlation between foot length and height (sitting).  

Result 5: Wk 14 Correlation (Height)   [Info]
Correlation matrix:
HeightSittingHeightUpperArmForearmHandUpperLegLowerLeg
SittingHeight0.64846383
(<0.0001)
UpperArm0.53501034
(0.0013)
0.14409236
(0.4237)
Forearm0.69535508
(<0.0001)
0.27913242
(0.1157)
0.47076814
(0.0057)
Hand0.58520461
(0.0003)
0.14830208
(0.4101)
0.64522081
(<0.0001)
0.5050472
(0.0027)
UpperLeg0.59470883
(0.0003)
0.18633213
(0.2992)
0.71597688
(<0.0001)
0.36582628
(0.0363)
0.60074246
(0.0002)
LowerLeg0.78294829
(<0.0001)
0.22635726
(0.2053)
0.66164552
(<0.0001)
0.7284308
(<0.0001)
0.54997862
(0.0009)
0.71497789
(<0.0001)
Foot0.49493199
(0.0034)
0.3680033
(0.0351)
0.14675004
(0.4151)
0.42765715
(0.013)
0.34707842
(0.0478)
-0.029770447
(0.8694)
0.28208099
(0.1117)

Multiple linear regression analysis was then performed for each dependent variable, first height (standing) and secondly for height (sitting). 

Result 6: Wk 14 Multiple Linear Regression (Height Standing)   [Info]
Multiple linear regression results:
Dependent Variable: Height
Independent Variable(s): UpperArm, Forearm, Hand, UpperLeg, LowerLeg, Foot

Parameter estimates:
ParameterEstimateStd. Err.DF95% L. Limit95% U. Limit
Intercept70.31509912.3409832644.94784695.682353
UpperArm-0.311050170.4658019426-1.26851980.64641943
Forearm0.774477120.6123497426-0.484225792.03318
Hand0.280315780.7847252326-1.332711.8933416
UpperLeg0.684378090.4205120526-0.17999681.548753
LowerLeg0.838869490.4956695226-0.17999381.8577328
Foot2.38689620.9429064260.448724354.3250681

Analysis of variance table for multiple regression model:
SourceDFSSMSF-statP-value
Model6568.1919394.69865412.44873<0.0001
Error26197.784447.6070937
Total32765.97636

Summary of fit:
Root MSE: 2.758096
R-squared: 0.7418
R-squared (adjusted): 0.6822
Residuals stored in new column: Residuals
Predicted values stored in new column: Pred. Values

Result 7: Wk 14 Multiple Linear Regression (Height Sitting)   [Info]
Multiple linear regression results:
Dependent Variable: SittingHeight
Independent Variable(s): UpperArm, Forearm, Hand, UpperLeg, LowerLeg, Foot

Parameter estimates:
ParameterEstimateStd. Err.DF95% L. Limit95% U. Limit
Intercept65.33122511.5794552641.52931589.133135
UpperArm-0.0862748040.4370585926-0.984661590.81211199
Forearm0.481444350.5745633226-0.699587471.6624762
Hand-0.655395650.73630226-2.16888610.85809478
UpperLeg0.539161030.3945634126-0.271875661.3501977
LowerLeg-0.294655110.4650831226-1.25064720.66133693
Foot1.80105750.8847222526-0.0175151713.6196301

Analysis of variance table for multiple regression model:
SourceDFSSMSF-statP-value
Model648.2452418.04087361.2006260.3372
Error26174.128096.6972343
Total32222.37333

Summary of fit:
Root MSE: 2.5879015
R-squared: 0.217
R-squared (adjusted): 0.0363
Residuals stored in new column: Residuals
Predicted values stored in new column: Pred. Values

It is appropriate to perform a "best fit" model, as when looking at our correlation matrix, there were several significant p-values and several that were not.  Therefore, a MLR was performed, using variable selection, where the p-values were set at 0.15 and 0.25.  This allows us to account for correlations among the predictor variables.  Using the "best fit" MLR model, the most significant correlations for height (standing) are:  lower leg lenght, foot length, and upper leg length.  For the height (sitting), the only significant correlation is with foot length.

Result 8: Wk 14 Multiple Linear Regression Height Standing w/variable selection   [Info]
Multiple linear regression results:
Dependent Variable: Height
Independent Variable(s): UpperArm, Forearm, Hand, UpperLeg, LowerLeg, Foot

Stepwise results:
P-value to enter: 0.15
P-value to leave: 0.25
StepVariableActionP-valueRMSER-squaredR-squared (adj)
1LowerLegEntered<0.00013.09227290.6130.6005
2FootEntered0.00822.79233310.69460.6743
3UpperLegEntered0.11842.72114220.71970.6907


Parameter estimates:
ParameterEstimateStd. Err.DF95% L. Limit95% U. Limit
Intercept77.45392510.502212955.97449498.933355
LowerLeg1.18969760.36455499290.444098911.9352963
Foot2.75481550.83924866291.03835934.4712718
UpperLeg0.524643280.3259808629-0.142062431.191349

Analysis of variance table for multiple regression model:
SourceDFSSMSF-statP-value
Model3551.24254183.7475124.81527<0.0001
Error29214.733827.4046146
Total32765.97636

Summary of fit:
Root MSE: 2.7211422
R-squared: 0.7197
R-squared (adjusted): 0.6907

Result 9: Wk 14 Multiple Linear Regression Height Sitting w/variable selection   [Info]
Multiple linear regression results:
Dependent Variable: SittingHeight
Independent Variable(s): UpperArm, Forearm, Hand, UpperLeg, LowerLeg, Foot

Stepwise results:
P-value to enter: 0.15
P-value to leave: 0.25
StepVariableActionP-valueRMSER-squaredR-squared (adj)
1FootEntered0.03512.49035630.13540.1075


Parameter estimates:
ParameterEstimateStd. Err.DF95% L. Limit95% U. Limit
Intercept76.3808024.19106433167.8330784.928534
Foot1.52404990.69162011310.113481422.9346184

Analysis of variance table for multiple regression model:
SourceDFSSMSF-statP-value
Model130.11522630.1152264.85582660.0351
Error31192.258116.2018744
Total32222.37333

Summary of fit:
Root MSE: 2.4903563
R-squared: 0.1354
R-squared (adjusted): 0.1075

Conclusions:

We can look at slope, but it is important to use caution when interpreting because of multicolinearity.

The significant predictors for height (standing) are: lower leg length, foot length, and upper leg length.  With 95% confidence, we have evidence that, at a fixed height (standing), every additional centimeter increase in Lower Leg results in an average of 0.44-1.94 additional centimeter increase in height (standing), every additional centimeter increase in foot length, results in an average of 1.04-4.47 additional centimeter increase in height (standing), and for every additional centimeter increase in upper leg length, results in an average of 0.14-1.19 additional centimeter increase in height (standing). 

The significant predictor for height (sitting) is foot length only.  With 95% confidence, we have evidence that every centimeter increase in foot length, results in an average of .11-2.93 additional centimeter increase in height (sitting).

The second part of our conclusion, is to estimate the heights of two subjects (women) when given measurements for upper arm length, forearm length, hand length, upper leg length, lower leg length, and foot length. (Measurements in cm are as follows)

Subject 1:  U-Arm=32, F-Arm=29, Hand=19, U-Leg=40, L-Leg=38, Foot=7

Subject 2:  U-Arm=34, F-Arm=30, Hand=19, U-Leg=44, L-Leg=42, Foot=6

Using the instructions for prediction estimation in StatCrunch (provided in the notes on p. 6 under "How to do PREDICTION"), the following results were obtained:

Subject 1:  With 95% confidence, we can predict that the average height (standing) is between 156.82 and 169.04 cm and her height (sitting) is between 81.71 and 92.38 cm.

Subject 2:  With 95% confidence, we can estimate that the height (standing) is between 161.33 cm and 172.74 cm and her height (sitting) is between 80.37 and 90.68 cm.

From our MLR analysis with variable selection, it would appear that both leg lengths and foot measurements are more significant predictors of height (standing), while foot length is the only significant predictor of height (sitting).

Limitations/Next Steps:

Limitations to this study is the small sample size--only 33 women were in the study.  Also, as environmental factors such as nutrition, exercise, and underlying health conditions can also contribute to height, these could be limitations and could possibly be evaluated in future studies.  It may also be beneficial to perform the study in males to look for a correlation between the measurements in males and between males and females.

 

HTML link:
<A href="https://www.statcrunch.com/5.0/viewreport.php?reportid=59218">Wk 15 Predicting Height--revision</A>

Comments
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By katie.dwyer20
Apr 30, 2016

Karen-
I am kind of glad that I did not do this report, because it all looks very confusing. There are so many graphs and analysis that I was a little intimidated on it all. Good job on your report.
-Kate
By nku.katie.waters
Apr 25, 2016

Hi Karen,
Nice job! Great job using stepwise regression. A couple comments:
1. The models you obtained in Results 7 and 8 are fine. If you compare your models with Jenna's, you'll see that she removed "Upper Leg" from the model for standing height. Either model is acceptable.

2. It is fine to interpret the slope as practice, but keep in mind that we must be cautious at interpreting slope because of multicollinearity. A correct interpretation would be: We have evidence that, at a fixed height (standing), every additional cm increase in LowerLeg results in an average of 0.44 to 1.94 additional cm increase in height (standing).

3. Your values for the prediction intervals are slightly off. It looks like you used a model containing all 6 variables for the prediction intervals instead of the model in result 8.

Let me know if you have any questions!
By jenna.dixon92
Apr 23, 2016

Great job!
I read your comments on my report, and I agree that Katie's comments would be a lot of help. This material has been pretty difficult for me to grasp. I noticed that in my Stepwise results (your Result 8) I removed Upper Leg from the table since its value wasn't significant.
By maggie.flowers
Apr 23, 2016

Posted my last comment on the wrong report...sorry. Your report looks great! I agree with you that the next study needs to include more people. 33 is a small number, and a larger sample would help to increase significance of the results.
By maggie.flowers
Apr 23, 2016

Good job on your report, I did the same one and there was a lot of information! Your report is well laid out and easy to follow.

Always Learning