STATISTICSCORNER Table 2: A partial listing of data available
STATISTICCORNER2 SarahWhite
70expectedheight
Sex
Age
height
Expectedheight
F
4
116
105.1
110'4
111.6
92',7
F
6
109
journal. welcome to the secondarticle in this new feature of the receive glad to be I would useful; one I hope you found the first
F
11
138
1,48.2
93 1
your comments.
F
14
r43
161.2
88 7
tl7
106.6
1098
The scenariowe consider in this issue is: pafticular You want to find out how the heights of children in a patterns growth normal for the population 'pipulation. compare with the is any there if know to like you would In particular age' with stunting, and if so if this changes
:
M4 M5
110
113.1
9 73
Mlz
t34
153
87 6
t56
171'5
91'o
M15 that we This question is really quite complex, since it requires the pop- Having obtainecl a standardised value how can we compare know what the normal expectedheights for children in Malawi' them with the expectedvalues ( I 00) ? ulation sampled is. Such norrns are not available for norms use is to Malawi in The usual approach of nutritionlists derived A useful method is to model how the values vary with age usinp provided in iables drawn up by the WHO' which are these univariate (also known as simple) linear regression' This type o1 irom populations in the US 1. Although this approachhas modelinvolvestwovariables,inthiscasetheageandth defecis it is the one used for the purposesof this article' exp ected heights of the children' sex) and (age, height To illustrate this situation lets use data betweer a Primary These models assumethat there is a linear telationship 'explanatory obtained from 64 children in standards I and 2 at the median the two variables. One variable is often called school in Mangochi district. The WHO tables provide 'explain'variation in the other vari inter- variable, since it is used to month one expectedheighis for children by sex and age,in 'response'variable only able. The other variable is usually called the uis. For the children sampled their ages were recorded thr anything This implies that there is a one-way relationship between using years. Thus a child recordedas age five could be variabl age of variables. In our scenario the age is the explanatory betweln five and six years minus one day' So the mean six while the 7o expectedheight is the response;the model assume and years children recorded as five years is actually five (c height to that the predicted Eo expected height of a child increases months. Therefore for each child we take the expected sex at six decreases)at a constant rate with age' be the median expected height of a child of that expected months beyond the number of years recorded' The cannot be exac l ' Table You may object that all children of the sameage in shown are data these for heishts used expectedheigh ty ttt" tu-"teight and therefore of the same Vo height expected 7o predicted Expectetl median heights for boys and girls aged concelns model itrat,s or, the valut actual 4 to 15 years old the that assuming by variation and allows for the Boys actt Girls between differences The Age values' predicted vary around the 'residuals'or 'errors'' mos yrs al and predictedvaluesare called 4
6
105.1
106.6
5
6
111.6
113.1
6
6
117.6
119.0
7
6
r23.5
r24.4
8
o
129.3
129.6
9
6
135.2
134.8
10
6
t41.5
t40.3
11
6
148.7
146.4
12
6
154.6
153.0
13
6
14
6
l)
6
r59.9 161.2
166.2 171.5
'standardised' by For each child hislher actual height can be expressingit as a percentageof the expectedheight' Hencefoith as these are called 7o expected heights (and also denoted final The 2' in Table shown is data ihe of Toexp-ht). A sample indicates how the child's height compareswith their'Vo "ololn expectedheight.
Three other assumptionsare also made: do the variability (standarddeviation) ofthe residuals ii) not change with age; distributt iii) this varitbility for any given age is normally age; that for value mean about the iv) the observationsare mutually independent;
as well as Epi In Statistical packagessuch as SPSSand Stata' to analyset SPSS used have I models' can be rrr"i to fii such the estim includes analysis regression a data. The output for in Table indicated is This (coefficients)' ed or fitted model 'Unstandardizedcoefficients/B'' Thus the I under the column ted model is: (1) Eaexp-ht=108.2-l'52*a8e What does this mean?
Thismeansthat,theheightofachild'asapercentage the age median expectedheights, is predicted by multiplying The ac 108'2' from amount this years by t.SZ anlsubtracting to id used be can model the but vary at vatues will of course popt the in children of heights expected tify the predicted Vo tion the samPlewas drawn from'
Malawi Medical Jo
34
STATISTICSCORNER Estimated Model Coefficients Unstandardized
Standardized
Coefficients
Cmfficients
B
Std. Error
Beta
Constant
108.16
2.55
AGE
-1.52
.21
Sig.
Confidence intenal Ilwer
Bomd
Upper Bomd
Constmt
.11.-19
.000
103.07
113.25
AGE
-5.6-;
.000
-2.06
-0.98
What does this tell us about how the heights compare with the ttonnal |alues? To answer this question lets begin by checking what the predicted 7c expectedheights are for children at the limits of the age range sampled,ie aged 4 and aged 15. Inserting these two values in equation (1) indicates that the expected values arc 1027o and857orespectively.This suggeststhat as children in this population increasein age they become increasingly shorter than the median expectedheight. In particular it is predicted that for two children differing in age by a year the older child's percent of the expectedheight is 1.5%,less than that of the younger child. This model gives us an estimateof the relationship by providing predicted Voexpectedheights,but is this relationship purely due to random variation or is it statistically significant? One approach to this question is to form a confidence interval for the slope of the line. We already have an estimate of the slope but to form a confidence interval we need to use the standard error of the estimate. In case you've forgotten a standard error of an estimateis in fact the estimatedstandarddeviation of the sampling distribution of the statistic, in this casethe statistic is the slope of the relationship being considered. This provides an indication of the accuracy and is given in Table 3. So for our examplethe estimateis -1.52, which can be thought of as being sampled from a Normal distribution with standard deviation 0.21. We can form a 957o confidenceinterval for the true value by taking an interval 1.96 (approximately 2) times this value above and below it. This interval (which is included in Table 3: -2.06,-0.98) indicates that we can be 957o certain thaL the slope of the relationship lies between 2 and -1. If there were no relationship then the actual slope would be 0.0, although the estimate is unlikely to be exactly zero. Since the estimated interval does not include zero we can conclude that there is a statistically significant relationship. In this caseas age increasesthe percentageof the expectedweight decreases. Summary Compared with a north American population the data for the children in our sample indicate that there is an increasing magnitude of stunting in Standard I and2 pupils in the population sampled.
Malawi Medical Joumal
The studentsused ranged in age from 4 to 15. It can be argued that if a l5-year old is only in Standard 2they arc likely to be less developedthan their peerswho are in Form 4, ie that the fact that they are in Standard i is related to poor nutritional status and hence development in stature. Thus it is not surprising that there is evidence in this datasetof stuntins. Furthermore, as indicated following the statement of the problem it is not known if this population is a suitable standardfor Malawi. It would be useful to develop standard tables for Malawi, although there are difficulties in doing so, arising from the practical issue of defining those eligible to include as members of a standardpopulation. Acknowledgements The data used were obtained by secondyenr studentsof the College of Medicine during their Community Health field work in July 2002. Invitation There are other questions to be asked about this analysis; these will be considered in the next Statistics Comer. You may like to suggest questions that you have for this data, or for other linear regression scenarios. I will be glad to receive your comments and questionson this or any statistical issue. Pleasesend them by e-mail to [email protected] or write to me at the Department of Community Health College of Medicine, Private Bag 360, Chichiri, Blantyre 3. Reference 1 Measuringchangein NutritionalStatus(detailsto be completed- I've askedDr Hofmanfor them).
STATISTICCORNER2 SarahWhite
70expectedheight
Sex
Age
height
Expectedheight
F
4
116
105.1
110'4
111.6
92',7
F
6
109
journal. welcome to the secondarticle in this new feature of the receive glad to be I would useful; one I hope you found the first
F
11
138
1,48.2
93 1
your comments.
F
14
r43
161.2
88 7
tl7
106.6
1098
The scenariowe consider in this issue is: pafticular You want to find out how the heights of children in a patterns growth normal for the population 'pipulation. compare with the is any there if know to like you would In particular age' with stunting, and if so if this changes
:
M4 M5
110
113.1
9 73
Mlz
t34
153
87 6
t56
171'5
91'o
M15 that we This question is really quite complex, since it requires the pop- Having obtainecl a standardised value how can we compare know what the normal expectedheights for children in Malawi' them with the expectedvalues ( I 00) ? ulation sampled is. Such norrns are not available for norms use is to Malawi in The usual approach of nutritionlists derived A useful method is to model how the values vary with age usinp provided in iables drawn up by the WHO' which are these univariate (also known as simple) linear regression' This type o1 irom populations in the US 1. Although this approachhas modelinvolvestwovariables,inthiscasetheageandth defecis it is the one used for the purposesof this article' exp ected heights of the children' sex) and (age, height To illustrate this situation lets use data betweer a Primary These models assumethat there is a linear telationship 'explanatory obtained from 64 children in standards I and 2 at the median the two variables. One variable is often called school in Mangochi district. The WHO tables provide 'explain'variation in the other vari inter- variable, since it is used to month one expectedheighis for children by sex and age,in 'response'variable only able. The other variable is usually called the uis. For the children sampled their ages were recorded thr anything This implies that there is a one-way relationship between using years. Thus a child recordedas age five could be variabl age of variables. In our scenario the age is the explanatory betweln five and six years minus one day' So the mean six while the 7o expectedheight is the response;the model assume and years children recorded as five years is actually five (c height to that the predicted Eo expected height of a child increases months. Therefore for each child we take the expected sex at six decreases)at a constant rate with age' be the median expected height of a child of that expected months beyond the number of years recorded' The cannot be exac l ' Table You may object that all children of the sameage in shown are data these for heishts used expectedheigh ty ttt" tu-"teight and therefore of the same Vo height expected 7o predicted Expectetl median heights for boys and girls aged concelns model itrat,s or, the valut actual 4 to 15 years old the that assuming by variation and allows for the Boys actt Girls between differences The Age values' predicted vary around the 'residuals'or 'errors'' mos yrs al and predictedvaluesare called 4
6
105.1
106.6
5
6
111.6
113.1
6
6
117.6
119.0
7
6
r23.5
r24.4
8
o
129.3
129.6
9
6
135.2
134.8
10
6
t41.5
t40.3
11
6
148.7
146.4
12
6
154.6
153.0
13
6
14
6
l)
6
r59.9 161.2
166.2 171.5
'standardised' by For each child hislher actual height can be expressingit as a percentageof the expectedheight' Hencefoith as these are called 7o expected heights (and also denoted final The 2' in Table shown is data ihe of Toexp-ht). A sample indicates how the child's height compareswith their'Vo "ololn expectedheight.
Three other assumptionsare also made: do the variability (standarddeviation) ofthe residuals ii) not change with age; distributt iii) this varitbility for any given age is normally age; that for value mean about the iv) the observationsare mutually independent;
as well as Epi In Statistical packagessuch as SPSSand Stata' to analyset SPSS used have I models' can be rrr"i to fii such the estim includes analysis regression a data. The output for in Table indicated is This (coefficients)' ed or fitted model 'Unstandardizedcoefficients/B'' Thus the I under the column ted model is: (1) Eaexp-ht=108.2-l'52*a8e What does this mean?
Thismeansthat,theheightofachild'asapercentage the age median expectedheights, is predicted by multiplying The ac 108'2' from amount this years by t.SZ anlsubtracting to id used be can model the but vary at vatues will of course popt the in children of heights expected tify the predicted Vo tion the samPlewas drawn from'
Malawi Medical Jo
34
STATISTICSCORNER Estimated Model Coefficients Unstandardized
Standardized
Coefficients
Cmfficients
B
Std. Error
Beta
Constant
108.16
2.55
AGE
-1.52
.21
Sig.
Confidence intenal Ilwer
Bomd
Upper Bomd
Constmt
.11.-19
.000
103.07
113.25
AGE
-5.6-;
.000
-2.06
-0.98
What does this tell us about how the heights compare with the ttonnal |alues? To answer this question lets begin by checking what the predicted 7c expectedheights are for children at the limits of the age range sampled,ie aged 4 and aged 15. Inserting these two values in equation (1) indicates that the expected values arc 1027o and857orespectively.This suggeststhat as children in this population increasein age they become increasingly shorter than the median expectedheight. In particular it is predicted that for two children differing in age by a year the older child's percent of the expectedheight is 1.5%,less than that of the younger child. This model gives us an estimateof the relationship by providing predicted Voexpectedheights,but is this relationship purely due to random variation or is it statistically significant? One approach to this question is to form a confidence interval for the slope of the line. We already have an estimate of the slope but to form a confidence interval we need to use the standard error of the estimate. In case you've forgotten a standard error of an estimateis in fact the estimatedstandarddeviation of the sampling distribution of the statistic, in this casethe statistic is the slope of the relationship being considered. This provides an indication of the accuracy and is given in Table 3. So for our examplethe estimateis -1.52, which can be thought of as being sampled from a Normal distribution with standard deviation 0.21. We can form a 957o confidenceinterval for the true value by taking an interval 1.96 (approximately 2) times this value above and below it. This interval (which is included in Table 3: -2.06,-0.98) indicates that we can be 957o certain thaL the slope of the relationship lies between 2 and -1. If there were no relationship then the actual slope would be 0.0, although the estimate is unlikely to be exactly zero. Since the estimated interval does not include zero we can conclude that there is a statistically significant relationship. In this caseas age increasesthe percentageof the expectedweight decreases. Summary Compared with a north American population the data for the children in our sample indicate that there is an increasing magnitude of stunting in Standard I and2 pupils in the population sampled.
Malawi Medical Joumal
The studentsused ranged in age from 4 to 15. It can be argued that if a l5-year old is only in Standard 2they arc likely to be less developedthan their peerswho are in Form 4, ie that the fact that they are in Standard i is related to poor nutritional status and hence development in stature. Thus it is not surprising that there is evidence in this datasetof stuntins. Furthermore, as indicated following the statement of the problem it is not known if this population is a suitable standardfor Malawi. It would be useful to develop standard tables for Malawi, although there are difficulties in doing so, arising from the practical issue of defining those eligible to include as members of a standardpopulation. Acknowledgements The data used were obtained by secondyenr studentsof the College of Medicine during their Community Health field work in July 2002. Invitation There are other questions to be asked about this analysis; these will be considered in the next Statistics Comer. You may like to suggest questions that you have for this data, or for other linear regression scenarios. I will be glad to receive your comments and questionson this or any statistical issue. Pleasesend them by e-mail to [email protected] or write to me at the Department of Community Health College of Medicine, Private Bag 360, Chichiri, Blantyre 3. Reference 1 Measuringchangein NutritionalStatus(detailsto be completed- I've askedDr Hofmanfor them).
