Answer to exercise 'height vs. age' (Juul)

Transcription

1 Answer to exercise 'height vs. age' (Juul) Question 1 Fitting a straight line to height for males in the age range 5-20 and making the corresponding illustration is performed by writing: proc reg data=juul; where age ge 5 and age le 20 and sex='male'; model height=age; proc gplot data=juul; where age ge 5 and age le 20 and sex='male'; plot height*age / haxis=axis1 vaxis=axis2 frame; axis1 value=(h=3) minor=none label=(h=3); axis2 value=(h=3) minor=none label=(a=90 R=0 H=3); symbol1 v=circle i=rl c=black h=2 l=1 w=2; It looks as if the variation increases somewhat with age, but for ease of interpretation, we abstain from performing any transformation. More importantly, it looks as if linearity is not entirely adequate, at least not extending beyond the age of 16. This is indeed not very surprising. 1

2 Question 2 Graphical ts of parabolas and cubic functions may be performed simply by using i=rq or i=rc in the symbol-statement in gplot. We get Quadratic ts: Cubic ts: In order to t the models for males and females simultaneously, we may use two dierent parametrisations. First, we choose the estimation friendy version: 2

3 proc glm data=juul; where age ge 5 and age le 20; class sex; model height=sex sex*age sex*age2 sex*age3 / noint solution; which gives The GLM Procedure Dependent Variable: height Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Uncorrected Total R-Square Coeff Var Root MSE height Mean Source DF Type I SS Mean Square F Value Pr > F sex <.0001 age*sex <.0001 age2*sex <.0001 age3*sex <.0001 Source DF Type III SS Mean Square F Value Pr > F sex <.0001 age*sex age2*sex <.0001 age3*sex <.0001 Standard Parameter Estimate Error t Value Pr > t sex female <.0001 sex male <.0001 age*sex female age*sex male age2*sex female age2*sex male <.0001 age3*sex female <.0001 age3*sex male <.0001 or with the test friendly parametrisation: 3

4 proc glm data=juul; where age ge 5 and age le 20; class sex; model height=age age2 age3 sex sex*age sex*age2 sex*age3 / solution; which gives The GLM Procedure Dependent Variable: height Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE height Mean Source DF Type I SS Mean Square F Value Pr > F age <.0001 age <.0001 age <.0001 sex <.0001 age*sex <.0001 age2*sex <.0001 age3*sex Source DF Type III SS Mean Square F Value Pr > F age age <.0001 age <.0001 sex age*sex age2*sex age3*sex Standard Parameter Estimate Error t Value Pr > t Intercept B <.0001 age B age B <.0001 age B <.0001 sex female B sex male B... 4

5 age*sex female B age*sex male B... age2*sex female B age2*sex male B... age3*sex female B age3*sex male B... NOTE: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable. We see that the two genders do not deviate in the coecient of the third order term (P=0.32), but (seen from Type I) they do dier in the coecient of the second order term (F=44.04, P < ). From the estimation friendly parametrisation, we may also note that the linear term is not signicant for girls (P=0.67), but this is rather hard to interpret... Question 3 Automatic t with e.g. i=sm40 gives the result which pretty much resembles the parabolic ts. Question 4. We now consider cutpoints at the ages 10, 12, 13 and 15 and dene variables 5

6 denoting number of years above these respective thresholds. data juul; set juul; /* definition af new intercept, at the age 5 */ age5=age-5; extra_age10=max(age-10,0); extra_age12=max(age-12,0); extra_age13=max(age-13,0); extra_age15=max(age-15,0); We now use these new variables to dene a linear spline, and at the same time we test linearity using a contrast-statement proc glm data=juul; where age ge 5 and age le 20; by sex; model lheight=age5 extra_age15 extra_age13 extra_age12 extra_age10 / solution; contrast 'all' extra_age10 1, extra_age12 1, extra_age13 1, extra_age15 1; From this we get sex=male The GLM Procedure Number of Observations Used 502 Dependent Variable: height Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE height Mean

7 Source DF Type I SS Mean Square F Value Pr > F age <.0001 extra_age <.0001 extra_age extra_age extra_age Source DF Type III SS Mean Square F Value Pr > F age <.0001 extra_age <.0001 extra_age extra_age extra_age Contrast DF Contrast SS Mean Square F Value Pr > F all <.0001 Standard Parameter Estimate Error t Value Pr > t Intercept <.0001 age <.0001 extra_age <.0001 extra_age extra_age extra_age Question 5 We nd the rst deviation from the straight line at the age of 13 (based on the Type I tests, P=0.0002). Question 6 For girls, we get similarly sex=female The GLM Procedure Number of Observations Used 628 Dependent Variable: height Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE height Mean

8 Source DF Type I SS Mean Square F Value Pr > F age <.0001 extra_age <.0001 extra_age <.0001 extra_age extra_age Source DF Type III SS Mean Square F Value Pr > F age <.0001 extra_age extra_age extra_age extra_age Contrast DF Contrast SS Mean Square F Value Pr > F all <.0001 Standard Parameter Estimate Error t Value Pr > t Intercept <.0001 age <.0001 extra_age extra_age extra_age extra_age and again, we see the rst deviation at the age of 13 (F=17.95, P < ). Question 7 A model for the two genders simultaneously is dened as proc glm data=juul; where age ge 5 and age le 20; class sex; model height=age5 extra_age10 extra_age12 extra_age13 extra_age15 sex sex*extra_age15 sex*extra_age13 sex*extra_age12 sex*extra_age10 sex*age5 / solution; and yields 8

9 The GLM Procedure Dependent Variable: height Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE height Mean Source DF Type I SS Mean Square F Value Pr > F age <.0001 extra_age <.0001 extra_age <.0001 extra_age <.0001 extra_age <.0001 sex <.0001 extra_age15*sex <.0001 extra_age13*sex <.0001 extra_age12*sex extra_age10*sex age5*sex Source DF Type III SS Mean Square F Value Pr > F age <.0001 extra_age extra_age extra_age extra_age <.0001 sex extra_age15*sex extra_age13*sex extra_age12*sex extra_age10*sex age5*sex Standard Parameter Estimate Error t Value Pr > t Intercept B <.0001 age B <.0001 extra_age B extra_age B extra_age B extra_age B <.0001 sex female B sex male B... extra_age15*sex female B extra_age15*sex male B... extra_age13*sex female B extra_age13*sex male B... 9

10 extra_age12*sex female B extra_age12*sex male B... extra_age10*sex female B extra_age10*sex male B... age5*sex female B age5*sex male B... NOTE: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable. We see some rather small signs of the genders dier already from the age of 10, although this does not reach signicance. From the age 13 and onwards, they clearly dier, since the boys grow more quickly than the girls. A scatter plot of the predictions looks like If we reduce the model to include only changes in the slope from the age 13 and onwards, we get The GLM Procedure Class Level Information Class Levels Values sex 2 female male Number of Observations Read 1142 Number of Observations Used 1130 Dependent Variable: height 10

11 Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE height Mean Source DF Type I SS Mean Square F Value Pr > F age <.0001 extra_age <.0001 extra_age <.0001 sex <.0001 extra_age15*sex <.0001 extra_age13*sex <.0001 Source DF Type III SS Mean Square F Value Pr > F age <.0001 extra_age extra_age <.0001 sex extra_age15*sex extra_age13*sex <.0001 Standard Parameter Estimate Error t Value Pr > t Intercept B <.0001 age <.0001 extra_age B <.0001 extra_age B <.0001 sex female B sex male B... extra_age15*sex female B extra_age15*sex male B... extra_age13*sex female B <.0001 extra_age13*sex male B... NOTE: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable. with a new plot of predicted values: 11

12 From the output as well as from the gures, we note that between the ages 13 and 15, boys grow an extra 4.4 cm (s.e.=0.62cm) a year on average, compared to the girls. Using the estimation friendly version of this model gives us Standard Parameter Estimate Error t Value Pr > t age <.0001 sex female <.0001 sex male <.0001 extra_age15*sex female extra_age15*sex male <.0001 extra_age13*sex female <.0001 extra_age13*sex male <.0001 Question 8 The Gompertz growth model takes the following form: height = α e β e γ age It is a bit tricky to determine starting values for this equation, since we have no possibility of transforming to linearity. First of all, we realize that when age increases, we approach an asymptote which is α. Hence, a reasonable guess of α is a maximum height, i.e. around 12

13 200cm. The parameter β is related to the timing of growth, whereas γ describes the growth velocity. Setting α to be 200, we may rearrange the above equation, and transform with the (natural) logarithm: height/200 e β e γ age log(200/height) β e γ age log(log(200/height)) log(β) γ age By dening the transformed variable y=log(log(200/height)), we may therefore estimate β and γ using linear regression, e.g. for males: data juul; set juul; y=log(log(200/height)); /* create rather crude starting values for nonlinear regression */ proc reg data=juul; model y=age; where age ge 5 and age le 20 and sex='male'; from which we get The REG Procedure Dependent Variable: y Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var

14 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 age <.0001 We therefore take the starting values to be: α = 200, β = exp( ) = 1.54, γ = Question 9 We now t the nonlinear model using these starting values proc nlin data=juul; where age ge 5 and age le 20; by sex; parms alfa=200 beta=1.5 gamma=0.15; model height=alfa*exp(-beta*exp(-gamma*age)); output out=ny p=yhat r=resid; sex=female The NLIN Procedure Dependent Variable height Approx Parameter Estimate Std Error Approximate 95% Confidence Limits alfa beta gamma sex=male The NLIN Procedure Approx Parameter Estimate Std Error Approximate 95% Confidence Limits alfa beta gamma

15 The output statement creates a new data set called ny, from which we may make residual plots, or as shown here, plots showing the tted relations: We note the slight increase in variance with age, which could be eliminated by a logarithmic transformation of height. Therefore we could instead t the model as data juul; set juul; lheight=log(height); proc nlin data=juul; where age ge 5 and age le 20; by sex; parms alfa=200 beta=1.5 gamma=0.15; model lheight=log(alfa)-beta*exp(-gamma*age); output out=ny p=yhat r=resid; 15

16 sex=female The NLIN Procedure Approx Parameter Estimate Std Error Approximate 95% Confidence Limits alfa beta gamma sex=male The NLIN Procedure Approx Parameter Estimate Std Error Approximate 95% Confidence Limits alfa beta gamma Question 10 We may compare males and females in various ways. From the estimates above, we may perform simple t-tests, e.g. for the parameter β: = 1.11 which is not signicant, no matter how many degrees of freedom this quantity might have (approximately). We may also perform a test in SAS by analysing the two genders simultaneously: proc nlin data=juul; where age ge 5 and age le 20; parms alfa=200 beta=1.5 gamma=0.15 alfadif=0 betadif=0 gammadif=0; if sex='male' then model lheight=log(alfa)-beta*exp(-gamma*age); if sex='female' then model lheight=log(alfa+alfadif)- (beta+betadif)*exp(-(gamma+gammadif)*age); 16

17 yielding The NLIN Procedure Dependent Variable lheight Approx Parameter Estimate Std Error Approximate 95% Confidence Limits alfa beta gamma alfadif betadif gammadif We may conclude that the αs and the γs dier between males and females, whereas the βs do not seem to dier. A cautious interpretation is that boys grow to become taller than girls (α) but that their growth appear to happen at a slower pace (γ). No dierence can be found in the timing of growth between the two genders. 17

18 Answer to additional questions for the BOC exercise Question 1 We t the nonlinear relation directly using the procedure nlin: proc nlin data=boc; parms beta=0.8 gamma=228; model boc=gamma*exp(-beta/days); output out=ny p=yhat r=resid; which creates the following output (plus some more, which is not shown here) Dependent Variable boc Method: Gauss-Newton Iterative Phase Sum of Iter beta gamma Squares NOTE: Convergence criterion met. NOTE: An intercept was not specified for this model. Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression <.0001 Residual Uncorrected Total Corrected Total

19 Approx Parameter Estimate Std Error Approximate 95% Confidence Limits beta gamma This provides a direct estimation of both parameters and we may compare the new condence intervals with the ones obtained from linearisation of the data: method β γ linearisation (0.727,0.888) (219.6,237.7) non-linear regression (0.713,0.952) (221.2,240.0) The dierences are not pronounced, but the nonlinear regression gives slightly wider condence intervals, especially for β. Question 2 The linear regression of the transformed relation is to be preferred since the assumption of variance homogeneity is better satised for this relation. This may also be seen from the residual plot below which is based on the nonlinear regression and which shows a clear 'trumpet' pattern. 19

20 However, the ts for the two models do not dier much as can be seen from the plots below: 20