Design effects associated with dietary nutrient intakes from a ... - Nature

European Journal of Clinical Nutrition (2007) 61, 1064–1071

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ORIGINAL ARTICLE

Design effects associated with dietary nutrient intakes from a clustered design of 1 to 14-year-old children PA Metcalf1,2, RKR Scragg2, AW Stewart2 and AJ Scott1 1 Department of Statistics, University of Auckland, Auckland 1, New Zealand and 2Department of Epidemiology and Biostatistics, University of Auckland, Auckland 1, New Zealand

Objective: To calculate intra-cluster and intra-household design effects and intra-class correlation coefficients for dietary nutrients obtained from a 24 h record-assisted recall. Design: Children were recruited using clustered probability sampling. Randomly selected starting-point addresses were obtained with probability proportional to mesh block size. Setting: Children aged 1–14 years in New Zealand. Subjects: There were 125 children in 50 clusters, giving an average of 2.498 children per cluster. In 15 homes, there were two children for the calculation of intra-household statistics. Results: Intra-cluster design effects ranged from 1.0 for cholesterol, b-carotene, vitamin A, vitamin D, vitamin E, selenium, fructose and both carbohydrate and protein expressed as their contribution to total energy intakes to 1.552 for saturated fat, with a median design effect of 1.148. Their corresponding intra-cluster correlations ranged from 0 to 0.37, respectively. Intrahousehold design effects ranged from 1.0 for height to 1.839 for vitamin B6, corresponding to intra-household correlations of 0 and 0.839. The median intra-household design effect was 1.550. Using a sampling design of two to three households per cluster for estimating dietary nutrient intakes would need, on average, a 15% increase in sample size compared with simple random sampling with a maximum increase of 55% to cover all nutrients. Conclusions: These data enable sample sizes for dietary nutrients to be estimated for both cluster and non-cluster sampling for children aged 1–14 years. The larger design effects found within households suggest that little extra information may be obtained by sampling more than one child per household. Sponsorship: The New Zealand Ministry of Health contracted this study.

European Journal of Clinical Nutrition (2007) 61, 1064–1071; doi:10.1038/sj.ejcn.1602618; published online 31 January 2007 Keywords: sample size; cluster sampling; between-cluster variation; dietary nutrients; children; adolescents

Introduction Cluster randomized trials or group sampling in which groups of subjects are allocated to different treatments are becoming increasingly popular (Campbell and Grimshaw, 1998). Correspondence: Dr PA Metcalf, Department of Community Health, University of Auckland, Private Bag 92019, Auckland 1, New Zealand. E-mail: [email protected] Guarantor: PA Metcalf. Contributors: RKRS, AWS and PAM contributed to the design of this study. PAM, AJS and AWS contributed to the statistical analysis and interpretation of the data. All authors contributed to the conduct of the study, data collection and writing of this manuscript. Received 1 November 2005; revised 16 November 2006; accepted 16 November 2006; published online 31 January 2007

Cluster sampling is an alternative to random sampling that retains the methodological strength of randomization (Donner et al., 1990; Koepsell et al., 1992). Examples of clusters are schools, neighbourhoods, suburbs, towns or cities. However, people in clusters cannot be treated as independent and the effect of this on an outcome leads to the need to increase the sample size (Campbell and Grimshaw, 1998; Kerry and Bland, 1998). This principle can also be applied to the recruitment of participants in epidemiological surveys. We applied this method for the collection of dietary information (24 h record-assisted recall and a qualitative food frequency questionnaire (FFQ)) in a pilot survey for the Children’s Nutrition Survey in New Zealand. This method

DEFs associated with dietary nutrient intakes of children aged 1–14 years PA Metcalf et al

1065 was considered appropriate as the study protocol required that an adequate sample of Pacific children be included by oversampling, and it is known that many Pacific people are not in local electoral rolls. Simple random sampling assumes that the data from each subject are independent of other subjects. Independence of data is a prerequisite for simple tests of significance such as Student’s t-tests and regression. However, when cluster sampling is used, individuals within each cluster are unlikely to be independent of each other because people in a cluster (e.g. a suburb) may share similar dietary attitudes, socioeconomic status and environmental factors, such as shopping in the same food shops. The consequence of these influences is that dietary nutrient intakes from individuals within clusters tend to be more similar to each other than dietary nutrient intakes from individuals from different clusters. As a result of this, the variance (or standard error) of between-group differences is usually larger than for the same number of randomly selected individuals, and its precision is usually less than that for the within-group variance. This reduces the statistical power and means that the number of subjects needs to be increased. Investigators need good estimates of the intra-class correlation coefficients for variables of interest, which together with the number of observations per cluster determines the size of the extra variation in the nested design. The design effect (DEF) (Kerry and Bland, 1998) or inflation factor (Reading et al., 2000) is the ratio of the variance using cluster randomization to the variance using individual randomization. It can be expressed in terms of the intra-cluster correlation (r) and the number in a single cluster, m: DEF ¼ 1 þ (m–1)r. If there is only one observation per cluster, m ¼ 1 then the DEF is 1.0, and the two designs are the same. Otherwise, the larger the intra-cluster correlation, that is, the larger the variation between clusters is, the bigger the DEF and more subjects will be needed to get the same power as a study which uses simple random sampling. For example, a DEF of 2.0 means that the sample size needs to be twice as large to give the same power as a simple random sample. The analysis of cluster randomized trials must also take into account the clustered nature of the data. Standard statistical techniques are not appropriate as they require data to be independent, unless aggregated analysis is performed at the level of the cluster. If the clustering effect is ignored, P-values will be artificially small, and confidence intervals will be too small, increasing the chances of spuriously significant findings and misleading conclusions. Techniques have now been developed to analyse data arising from a clustered design, which allow the hierarchical nature of the data to be modelled appropriately (Rice and Leyland, 1996). They allow variation to be modelled at each level of the data, e.g., at both the cluster and the household levels. The main difficulty in calculating sample sizes for cluster sample studies is in obtaining an estimate of the withincluster variation or intra-cluster correlation, as these do not

appear to have been published widely in the past (Campbell and Grimshaw, 1998). We conducted a pilot study using cluster sampling of children to assess their dietary nutrient intakes. Here, we report both intra-cluster and intra-household DEFs, and intra-class correlation coefficients for a number of nutrient intakes, height, weight and haemoglobin associated with randomizing street addresses rather than individuals. This information is needed for calculating required sample sizes of children’s nutrition surveys using clustered sampling.

Methods The aim of this study was to pilot sampling methods for a national Children’s Nutrition Survey by collecting dietary and other information from a clustered random sample of children in Auckland, Shannon and Feilding. All aspects of the study had ethical approval from the Auckland Ethics Committee.

Recruitment of children A total of 125 children in Auckland and Shannon were recruited in the pilot testing of methods (response rate 70%). The pilot testing was to assess the acceptability of the 24-h record-assisted recall computerized programme, anthropometric measurements, socio-demographic, physical activity, food security and medical history questions, and the repeatability of the FFQ (Metcalf et al., 2003). Starting points (addresses) were randomly selected with probability proportional to mesh block size in prescribed areas by Statistics New Zealand. Recruiters went to the starting point address and then visited the next 19 houses to the left of the starting point address. From these houses, 125 children were selected according to the following probabilities: Maori, 0.32; Pacific, 0.79; and Other, 0.32. These probabilities came from the proportion of children expected in each ethnic group in the population under study. Nutrients were calculated using the New Zealand Food Composition Database (New Zealand Food Composition Database, OCNZ93).

Statistical analysis Statistical analyses were performed with the Statistical Analysis System (SAS Institute Inc., 2004). Median nutrient intakes are reported, and standard deviations were calculated using a very robust-scale estimator, the median absolute deviation (MAD) about the median (Hampel, 1974), which is an option in the SAS PROC UNIVARIATE procedure (SAS Institute Inc., 2004). SAS PROC MIXED was used to estimate these variance components using restricted maximum likelihood (REML) using the following SAS code: PROC MIXED; CLASS cluster; MODEL nutrient ¼ RANDOM cluster; which results in a cluster variance component – the between-cluster European Journal of Clinical Nutrition


1066 Table 1 Median (MAD) daily dietary nutrient intakes, weight, height and haemoglobin levels in 40 children aged 1–4 years, 44 children aged 5–9 years and 41 children aged 10–14 years Variable Energy (MJ) Carbohydrate (g) Carbohydrate as % energy Protein (g) Protein as % energy Fat (g) Fat as % energy Saturated fat (g) Saturated fat as % energy Monounsaturated fat (g) Monounsaturated fat as % energy Polyunsaturated fat (g) Polyunsaturated fat as % energy Water (g) Fibre (g) Sugars (g) Glucose (g) Fructose (g) Sucrose (g) Lactose (g) Maltose (g) Starch (g) Cholesterol (mg) Thiamine (mg) Riboflavin (mg) Niacin total (mg) Niacin equivalents from tryptophan (mg) Vitamin A (mg) Vitamin C (mg) Vitamin D (mg) Vitamin E (mg) Vitamin B6 (mg) Vitamin B12 (mg) Folate (mg) Beta carotene (mg) Retinol (mg) Sodium (mg) Potassium (mg) Magnesium (mg) Calcium (mg) Phosphorus (mg) Iron (mg) Zinc (mg) Manganese (mg) Copper (mg) Selenium (mg) Weight (kg) Height (mm) Haemoglobin (g/l)

1–4 (years)

5–9 years

5.766 (2.266) 195 (71.8) 54.5 (9.95) 49.7 (23.0) 13.4 (4.10) 47.6 (25.3) 31.7 (5.98) 20.7(12.37) 14.4 (4.46) 16.1 (9.28) 9.1 (3.18) 6.5 (3.77) 4.0 (1.84) 1285 (432.4) 11.6 (5.33) 113.7 (75.7) 15.6 (10.0) 17.5 (12.6) 47.4 (41.4) 14.3 (9.46) 2.46 (1.68) 88.1 (39.0) 123.0 (70.7) 1.39 (0.65) 1.55 (0.67) 9.5 (4.04) 20.1 (7.34) 486.4 (305.8) 81.7 (77.2) 1.70 (1.68) 4.56 (2.32) 1.00 (0.74) 2.65 (1.82) 149.2 (56.4) 668.7 (658.0) 264.8 (137.0) 1615 (560.5) 2127 (992.6) 198.4 (77.5) 627.2 (326.4) 851.8 (331.5) 7.7 (2.94) 6.4 (2.42) 1892 (667.7) 0.75 (0.45) 22.7 (12.23) 15.5 (2.52) 960.7 (108.5) 125.0 (4.45)

7.901 243 50.3 62.4 12.6 82.8 32.1 32.1 13.8 26.3 11.1 8.0 4.1 1347 14.3 116.0 15.5 17.1 62.7 13.9 3.20 116.5 188.4 1.42 1.63 12.7 27.8 582.5 104.2 1.73 6.37 1.19 2.61 170.7 1055.6 329.4 2520 2164 221.2 599.5 1109.9 9.8 8.2 2423 1.01 24.3 23.9 1248.0 129.0

(2.487) (73.2) (7.34) (27.0) (4.24) (29.3) (7.98) (18.97) (4.82) (13.33) (3.34) (4.70) (1.56) (507.5) (4.08) (55.5) (9.0) (15.1) (45.9) (7.74) (1.98) (39.1) (110.6) (0.83) (0.66) (4.03) (11.69) (409.6) (105.9) (1.31) (3.05) (0.65) (1.63) (68.2) (996.6) (225.0) (1257.1) (670.6) (57.8) (347.9) (414.9) (2.66) (3.10) (851.2) (0.42) (9.29) (7.27) (106.0) (5.93)

10–14 years 8.674 247 52.1 73.6 14.2 86.1 35.6 37.0 15.0 28.8 11.5 10.0 3.6 1470 17.5 118.6 14.2 18.1 59.4 14.0 3.95 129.5 256.4 1.79 1.75 14.7 29.3 627.4 114.1 2.71 7.37 1.21 4.05 184.9 1015.0 257.2 2722 2501 226.9 649.6 1177.5 11.2 10.7 2688 1.01 38.6 50.0 1543.5 133.0

(2.963) (96.2) (7.35) (33.5) (3.77) (27.2) (6.07) (17.75) (3.55) (10.50) (3.02) (6.24) (1.76) (402.3) (7.88) (60.5) (12.5) (15.5) (34.5) (11.43) (3.63) (58.9) (171.4) (1.08) (0.77) (6.92) (12.24) (375.4) (100.7) (2.32) (4.21) (0.61) (2.66) (67.6) (835.9) (207.3) (1052.3) (961.0) (83.8) (457.7) (458.6) (3.97) (6.43) (1140.1) (0.47) (23.25) (13.64) (127.6) (5.93)

Abbreviation: MAD, median absolute deviation is a robust measure of the standard deviation.

variance s2c and a residual variance component – the withincluster variance s2r (Gulliford et al., 1999) from the same children. For each variable, model assumptions were checked by plotting the residuals against their predicted values for the fixed and random effects and examining a normal plot of the residuals, and examining a normal plot of the cluster estimates of the random effects where the cluster variance was 40 (as the predicted values are a constant). Outliers were European Journal of Clinical Nutrition

removed, if necessary, and the models refitted to see if the results changed substantially. However, it has previously been noted that the PROC MIXED estimators of the cluster and residual variance derived under normality assumptions are reasonable estimators even when their distributions are unspecified (Harville, 1977). The intra-class correlation s2c coefficient (ri) was then calculated as ðs2 þs 2 Þ (Gulliford et al., c r 1999). DEFs were calculated using the following formula:


1067 Table 2 Within-cluster design effects, intraclass correlation coefficients and variance components for dietary nutrients, weight, height and haemoglobin from 125 children Nutrient Energy (MJ) Carbohydrate (g) Carbohydrate as % energy Protein (g) Protein as % energy Fat (g) Fat as % energy Saturated fat Saturated fat as % energy Monounsaturated fat (g) Monounsaturated fat as % energy Polyunsaturated fat (g) Polyunsaturated fat as % energy Water (g) Fibre (g) Sugars (g) Glucose (g) Fructose (g) Sucrose (g) Lactose (g) Maltose (g) Starch (g) Cholesterol (mg) Thiamine (mg) Riboflavin (mg) Niacin total (mg) Niacin equivalents from tryptophan (mg) Vitamin A (mg) Vitamin C (mg) Vitamin D (mg) Vitamin E (mg) Vitamin B6 (mg) Vitamin B12 (mg) Folate (mg) Beta carotene (mg) Retinol (mg) Sodium (mg) Potassium (mg) Magnesium (mg) Calcium (mg) Phosphorus (mg) Iron (mg) Zinc (mg) Manganese (mg) Copper (mg) Selenium (mg) Weight (kg) Height (mm) Haemoglobin (g/l)

Between cluster variance

Within cluster variance

Intraclass correlation coefficient

Design effect

3.813 2753.51 0.00 211.22 0.00 467.37 2.39 118.48 2.27 51.81 0.17 0.70 0.62 28 469.00 4.12 759.96 9.12 0.00 236.78 5.29 0.79 80.85 0.00 0.22 0.14 6.17 68.00 0.00 650.26 0.00 0.00 0.16 1.09 1260.40 0.00 2176.86 214 497.00 215 649.00 3338.64 15 845.00 66 850.00 9.49 4.39 388 561.00 4.60 0.00 12.77 10 542.00 14.91

8.077 11 013.00 90.44 629.96 17.02 1031.64 58.88 203.23 16.56 140.22 10.40 48.24 6.08 221 417.00 122.37 5660.00 139.77 178.36 4698.69 136.77 8.99 4935.27 28 124.00 1.27 0.85 31.58 126.09 11 106 419.00 14 690.00 5.10 18.45 1.97 4.62 8746.57 402 990 000.00 56 721.00 1 495 629.00 1 044 789.00 10 464.00 133 872.00 142 527.00 18.53 21.23 1 764 106.00 112.19 579.29 295.83 666 592.00 66.64

0.321 0.200 0.000 0.251 0.000 0.312 0.039 0.368 0.120 0.270 0.016 0.014 0.092 0.114 0.033 0.118 0.061 0.000 0.048 0.037 0.081 0.016 0.000 0.149 0.142 0.163 0.350 0.000 0.042 0.000 0.000 0.077 0.191 0.126 0.000 0.037 0.125 0.171 0.242 0.106 0.319 0.339 0.171 0.181 0.039 0.000 0.041 0.016 0.183

1.480 1.300 1.000 1.376 1.000 1.467 1.059 1.552 1.180 1.404 1.024 1.022 1.138 1.171 1.049 1.177 1.092 1.000 1.072 1.056 1.121 1.024 1.000 1.223 1.212 1.245 1.525 1.000 1.064 1.000 1.000 1.115 1.286 1.189 1.000 1.055 1.188 1.256 1.362 1.159 1.478 1.507 1.257 1.270 1.059 1.000 1.062 1.023 1.274

DEF ¼ 1 þ (m01) ri (Gulliford et al., 1999). The mean cluster size was calculated as (Armitage and Berry, 1994): 13 0 k P m2i C7 6 B 1 6 Bi¼1 C7 m0 ¼ 6n B C7 @ n A5 ðc 1Þ 4 2

where c is the total number of clusters, mi is the number of children in the ith cluster and n is the total number of

children in the sample. In the 15 households with two children sampled, the second child was excluded from the within-cluster calculations.

Results Children were aged 1–14 years with a median of 7 years. There were 49 girls and 76 boys. Table 1 shows medians, MADs (a robust measure of the standard deviations) for daily European Journal of Clinical Nutrition


1068 nutrient intakes, weight, height and haemoglobin levels by age groups 1–4, 5–9 and 10–14 years. As expected, in general, median nutrient intakes, weight and height increased with age. Exceptions were carbohydrate, protein, saturated fat, all expressed as their percentage contribution to total energy intakes, glucose, fructose, sucrose, lactose, beta carotene, retinol, calcium and copper. DEFs were calculated for sampling by cluster (intra-cluster) and for sampling more than one child per household (intrahousehold).

Intra-cluster design effects There were 125 children in the random sample of 50 clusters, giving an average of 2.498 children per cluster, with a range of 1–6. The variance components, DEF and intra-cluster correlation for individual nutrients, weight, height and haemoglobin are reported in Table 2. Dietary nutrient DEFs ranged from 1.0 for cholesterol, beta carotene, vitamin A, vitamin D, vitamin E, fructose, and both carbohydrate and protein when expressed by their contribution to total energy intakes to 1.552 for saturated fat, with a median DEF of 1.148. Their corresponding intra-cluster correlation coefficients ranged from 0.00 to 0.37, respectively. The DEFs (and corresponding intra-cluster correlations (ri)) for weight (1.062; ri ¼ 0.041), height (1.023; ri ¼ 0.016) and haemoglobin (1.273; ri ¼ 0.183) were all relatively small. DEFs for nutrients were categorized into groups of width 0.1 to determine whether the number of DEFs in each category decreased with increasing size of the effect, which could suggest random outliers. Figure 1 shows the frequency distribution of the DEFs across the 41 crude nutrients. Nutrients expressed as their total contribution to total

energy intakes were not included in this figure because they are a derived quantity and because some nutrients would have been included twice. As the frequency of the size of the DEFs did not decrease much as the DEFs got larger, this suggested that there were important DEFs within clusters. Note that the size of the between-cluster and withincluster variance components is dependent on the magnitude of the nutrient intakes.

Intra-household design effects There were 30 children from 15 households, with two children per household. The DEF and intra-household correlation for individual nutrients, weight, height and haemoglobin are given in Table 3. Intra-household DEFs ranged from 1 for height to 1.839 for vitamin B6, corresponding to intra-household correlations of 0 and 0.839, respectively. The median within-household DEF was 1.550. In general, the DEFs were much larger within households than within clusters. Exceptions were for fat expressed by its contribution to total energy intake, manganese, height and haemoglobin.

Example The following example shows how the data in Tables 1 and 2 can be used for calculating sample sizes. To detect a difference in total energy of 0.500 MJ between two groups of 1–4-year-old children at the 5% significance level and 80% power, then n¼

2 2 Za=2 þ Z1b s2 D2

where n is the number of people required per group, Z is from the normal distribution, a is the significance level, b is the power, s2 is the standard deviation squared, and D is the difference between groups to be detected. If we use a random sample n ¼ 2 (1.96 þ 0.84)2 (2.266)2/ 0.5002 ¼ 324 children per group, and allowing for the DEF for a clustered sample, we have n ¼ 1.492 324 ¼ 484 children per group.

Discussion We have reported both intra-cluster and intra-household DEFs, and intra-class correlation coefficients calculated from 24-h-assisted recall data for a number of nutrient intakes, height, weight and haemoglobin, associated with randomizing street addresses rather than individuals.

Figure 1 Frequency of intra-cluster DEFs for nutrients categorized into groups of width 0.1.

European Journal of Clinical Nutrition

Recruitment method The sampling scheme in the current study used probabilities of selection for the different ethnic groups in the area under


1069 Table 3 Within-household design effects, intraclass correlation coefficients and variance components for nutrients, weight, height and haemoglobin from 30 children Nutrient Energy (MJ) Carbohydrate (g) Carbohydrate as % energy Protein (g) Protein as % energy Fat (g) Fat as % energy Saturated fat (g) Saturated fat as % energy Monounsaturated fat (g) Monounsaturated fat as % energy Polyunsaturated fat (g) Polyunsaturated fat as % energy Water (g) Fibre (g) Sugars (g) Glucose (g) Fructose (g) Sucrose (g) Lactose (g) Maltose (g) Starch (g) Cholesterol (mg) Thiamine (mg) Riboflavin (mg) Niacin total (mg) Niacin equivs from tryptophan (mg) Vitamin A (mg) Vitamin C (mg) Vitamin D (mg) Vitamin E (mg) Vitamin B6 (mg) Vitamin B12 (mg) Folate (mg) Beta carotene (mg) Retinol (mg) Sodium (mg) Potassium (mg) Magnesium (mg) Calcium (mg) Phosphorus (mg) Iron (mg) Zinc (mg) Manganese (mg) Copper (mg) Selenium (mg) Weight (kg) Height (mm) Haemoglobin (g/l)

Between household variance

Within household variance

Intraclass correlation coefficient

Design effect

9.44 6679.28 14.07 528.96 4.03 1310.27 16.91 335.66 8.92 192.05 4.63 21.95 7.25 153 284.00 36.67 3274.66 175.60 155.65 567.81 73.19 7.94 1035.05 11 374.00 1.39 0.70 19.57 108.08 98 029.00 10 290.00 1.68 6.74 0.64 4.79 4151.76 921 447.00 43 001.00 1 078 200.00 1 233 911.00 7821.54 112 323.00 152 321.00 31.75 15.69 376 605.00 0.21 191.74 16.94 0.00 19.24

9.77 10 412.00 59.53 430.55 10.81 1223.09 31.43 326.00 10.84 123.04 5.02 14.07 2.49 107 859.00 40.89 2393.33 45.63 66.13 1969.88 157.78 6.18 3604.03 7653.16 0.71 0.41 20.81 74.95 164 901.00 4517.86 3.93 8.77 0.12 3.82 7207.92 2 773 847.00 58 937.00 786 317.00 405 616.00 3605.88 84 900.00 124 910.00 11.05 9.67 1 138 182.00 0.06 149.46 241.92 93 868.00 105.75

0.492 0.391 0.191 0.551 0.271 0.517 0.250 0.507 0.451 0.610 0.479 0.609 0.744 0.587 0.473 0.578 0.794 0.702 0.224 0.317 0.562 0.223 0.598 0.661 0.630 0.485 0.591 0.373 0.695 0.299 0.435 0.839 0.556 0.365 0.249 0.422 0.578 0.753 0.684 0.570 0.549 0.742 0.619 0.249 0.789 0.562 0.065 0.000 0.155

1.492 1.391 1.191 1.551 1.271 1.517 1.350 1.507 1.451 1.610 1.479 1.609 1.744 1.587 1.473 1.578 1.794 1.702 1.224 1.317 1.562 1.223 1.598 1.661 1.630 1.485 1.591 1.373 1.695 1.299 1.435 1.839 1.556 1.365 1.249 1.422 1.578 1.753 1.684 1.570 1.549 1.742 1.619 1.249 1.789 1.562 1.065 1.000 1.155

study, which meant that there was only a probability of 0.24 of two children being recruited from any one household, and a very small probability (o0.008) of more than two children being selected from a single household. It was also possible that no children from a household with eligible children were selected. The advantage of this selection process was that, within the three ethnic groups, all individuals were selected with the same probability and hence no weighting would be required for within-group analyses. In addition, the selection probabilities were such that all three ethnic

groups would have approximately the same number of children selected in the final sample.

Intra-cluster DEFs The DEFs and intra-cluster correlations reported in Table 2 suggest that the method of cluster sampling used in this pilot survey would increase the required sample size by 15% on average for most nutrients, with a maximum required European Journal of Clinical Nutrition


1070 increase of 55% to cover all nutrients, compared to using simple random sampling. For example, the intra-cluster correlation of 0.32 for total energy means that for a sample size of 20 per cluster, the DEF is 1 þ 19 0.321 ¼ 7.10. Thus, there is a cost associated with larger-sized clusters. This ‘cost’ depends on the relative costs of recruiting and interviewing. Connelly (2003) describes how to select economically efficient combination(s) of clusters and cluster size.

Intra-household design effects Table 3 shows that the DEFs were generally larger within a household than within clusters. This shows that the diet of children living in the same household were much more similar than children living in other houses. The main reason for estimating the within-household DEFs was to determine the degree to which the DEFs were inflated. The two main factors that influence this are (i) the intra-household correlation. If this is high, it means the characteristics (e.g. nutrient intakes) of children within the same household are very similar. This suggests that only one child should be sampled per household for a study similar to the current one (in effect, sampling the other children adds little further information to the survey results because of the close similarities of dietary intakes within a household). However, if the analysis of the data focuses on examining age groups separately, any DEFs introduced by sampling more than one per household would be reduced in such an analysis because the children within a household are likely to be spread across different age groups, and (ii) variability in the household sizes. Sampling only one per household leads to disproportionately sampling children who belong to households with fewer children. This does not necessarily lead to any bias (any potential bias is removed by the use of weights inversely proportional to the household size), but it is not necessarily very efficient (especially as the Pacific Island population was a key subdomain of interest). This population group has more variably sized households (with regard to the number of children in New Zealand). An additional factor that needs to be taken into account relates to the practical issues surrounding the fieldwork for the survey. Reducing the costs of interviewers’ travel and time is a prime factor in undertaking a clustered approach. Sampling more children per household would mean less travel, less work making contact with households and less work convincing parents to participate. Once one child has been surveyed, the marginal costs of including further children from the same household would most likely be considerably smaller than moving to a different household to survey the next child. The decision of whether to select more than one child in a household depends on the relative costs of interviewing more than one child in a household versus the costs of going to single households. For example, the intra-household correlation of 0.492 for total energy intake means that collecting information from up to three children in a European Journal of Clinical Nutrition

household, the DEF is 1 þ 2 0.49 ¼ 1.984. In contrast, the intra-household correlation for copper of 0.789 gives a DEF of 2.578, giving very little additional information for the extra two children interviewed. Thus, in general, withinhousehold variation is smaller than within-cluster variation. An inverse relationship between-cluster size and the degree of within-cluster variation has been described previously (Gulliford et al., 1999). Although there is a need for the publication of variance components and intra-class correlation coefficients to aid the design of complex surveys, these do not appear to have been published widely in the past (Campbell and Grimshaw, 1998). A selection of studies that have done can be found in the book by Donner et al. (Donner et al., 2000). The intraclass correlation coefficient is more generalizable than the DEF, as the latter is dependent on cluster size. These data enable sample sizes to be estimated for cluster sampling of dietary nutrient intakes in children aged 1–14 years. The larger DEFs found within households suggest that little extra information may be obtained by sampling more than one child per household, but this decision should be governed by the relative costs of interviewing more than one child per household versus the costs of going to single households.

Acknowledgements Other collaborators on the Children’s Nutrition Pilot Survey were Professor Boyd Swinburn, Dr Cameron Grant and Dr David Schaaf from the University of Auckland, Professor Mason Durie and Eljon Fitzgerald (Massey University, Palmerston North), Dr Elaine Rush (Auckland University of Technology) and Dr Clare Wall, Kate Sladden and Patsy Watson (Massey University, Albany). Dr Patricia Metcalf was supported by the Health Research Council of New Zealand.

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