This survey was conducted to find what the forecasters of Body Mass Index are. There were two research inquiries of this survey. First research inquiry was How good the type of cocoa and frequence of cocoa ingestion predict organic structure mass index, after commanding for gender physical activity? Second research inquiry was “ How good do fat per centum and chocolate tree per centum in cocoa explain organic structure mass index, after commanding the consequences of the first research inquiry? ” In order to uncover the forecasters hierarchal arrested development analysis was used. In this survey BMI was outcome variable ; gender, type of cocoa, fat rate in cocoa, chocolate rate in cocoa, frequence of cocoa ingestion and frequence of physical activity in a hebdomad were predictor variables. The survey was conducted with 600 university pupils.
Method
Participants and the Variables
The sample of the survey was consisted of 600 Middle East Technical University pupils ; 46.3 % ( n=278 ) were male and 53.7 % ( n=322 ) were female. Convenience trying method was used to find the participants. The most crowded topographic points of the university, such as library, market country, dormitory country, were selected as informations aggregation countries.
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Needed sample size for multiple arrested development could be calculated with the expression of figure of forecasters * 8 + 50. Harmonizing to expression required sample size is 106 ( 7*8+50 ) . While there are 600 pupils, sample size is rather adequate to carry on multiple arrested development.
The questionnaire used in this survey was consisted of seven points which are presented in Table 1. Furthermore, there is an id figure for each participant. Wholly, there were six uninterrupted and two categorical variables on informations file.
Table 1
List of variables and brief descriptions in the informations file
Variable Name
Description of the variable
Gem state
Identity figure of each participant
Body mass index
Body Mass Index
Gender
Gender ( 1: Male ; 2: Female )
Type
Type of cocoa ( 1: Milk ; 2: Berry ; 3: Peanut )
Fat
Fat rate ( % ) in cocoa
Cacao
Cacao rate ( % ) in cocoa
Frequency
Frequency of cocoa ingestion ( figure of cocoas eaten in the last hebdomad )
Activity
Frequency of physical activity in a hebdomad
Datas Analysis Plan
In this survey hierarchal arrested development will be held to happen out how much the forecasters can explicate the dependant variable, BMI. In hierarchal arrested development different theoretical accounts are tested consecutive. In contrast to stepwise arrested development, research worker decides the sequence of the forecasters that included the theoretical account.
Three different theoretical accounts will be used to find how much these independent variables predict the dependant variable. In the first theoretical account gender and frequence of physical activity in a hebdomad will be included into analysis. In the 2nd theoretical account, gender and frequence of physical activity in a hebdomad will be controlled ; type of cocoa and frequence of cocoa ingestion will be included into analysis. In the 3rd theoretical account, gender, frequence of physical activity in a hebdomad, type of cocoa and frequence of cocoa ingestion will be controlled, fat per centum and chocolate tree per centum in cocoa will be included into analysis.
To carry on the arrested development analysis, categorical informations should be recoded. There are three different ways to make this ; silent person cryptography, effects coding and contrast cryptography. In this survey, silent person cryptography will be used to recode categorical informations. In dummy cryptography, one categorical variable recode into different variables that the figure of new variables are one less than the figure of classs. Nevertheless, a categorical variable should hold at least three degrees to be recoded. A categorical variable with two degrees such as gender need n’t to be recoded. In this survey there were two categorical informations ; gender and type of cocoa. As it mentioned before, gender need n’t to be recoded. The other categorical variable, type of cocoa, should be recoded. Milk cocoa will be selected as mention variable ; and, two other variables will be coded as milkvsberry and milkvspeanut.
Similarly all other multivariate statistical methods, Multiple Regression has assorted premises ; and, all these premises should be checked before carry oning the analysis. First premise of multiple arrested development is normality. Unlike other multivariate analysis, arrested development analysis checks whether the mistake distributes usually or non. Second, multicollinearity, which is high degree of intercorrelation among forecaster variables, should be checked. Third, premise of homoscedasticity should be checked. Homoscedasticity assumes that the discrepancy of the error term is changeless across each value of the forecaster. This means that there should non be seen a form on spread secret plan. Fourth premise is independency, that the error term is independent of the forecasters in the theoretical account and of the values of the error term for other instances. The 5th premise of multiple arrested development is one-dimensionality. Last, outliers should be look into whether they affect the consequences or non. Partial secret plans, purchase statistics, Cook ‘s D, DFBeta and Mahalonobis distance could be used to find outliers.
Consequences
Descriptive Statisticss
Table 2 shows the descriptive statistics of the survey. Table 2 shows that there is no losing informations ; mean of dependant variable, BMI, is 24.65 and the standard divergence is 4.48.
Table 2
Descriptive Statisticss
Mean
Std. Deviation
Nitrogen
organic structure mass index
24.65
4.48
600
Gender
1.54
.50
600
physical activity in a hebdomad
2.62
.74
600
milk cocoa V berry cocoa
.25
.44
600
milk cocoa V peanut cocoa
.27
.45
600
frequence of cocoa ingestion
4.66
.73
600
fat rate ( % ) in cocoa
51.70
9.69
600
chocolate tree rate ( % ) in cocoa
51.95
9.96
600
Table 3 shows the correlativities between the variables. If the tabular array is examine it is seen that the best forecaster of BMI is fat rate in cocoa. There is a positive and high correlativity between the BMI and fat rate in cocoa. On the other manus, there is no correlativity between BMI and gender, physical activity in a hebdomad, milk cocoa V berry cocoa. Furthermore, there is no correlativity higher than.90 between the independent variables.
Table 3
Correlation Matrix
1
2
3
4
5
6
7
8
Pearson Correlation
organic structure mass index ( 1 )
1.00
Gender ( 2 )
-.03
1.00
physical activity in a hebdomad ( 3 )
.04
-.13
1.00
milk cocoa V berry cocoa ( 4 )
-.03
.03
-.11
1.00
milk cocoa V peanut cocoa ( 5 )
.23
-.02
.12
-.36
1.00
frequence of cocoa ( 6 ) ingestion
.31
.12
.15
-.05
.19
1.00
fat rate ( % ) in cocoa ( 7 )
.64
-.12
.08
.02
.21
.30
1.00
chocolate tree rate ( % ) in cocoa ( 8 )
.52
.08
.03
-.04
twenty-two
.28
.51
1.00
Premises
The first premise of multiple arrested development to be checked is normalcy. Unlike other analysis, normalcy of remainders is checkered whether mistakes usually distributed or non. Normality of remainders could be checked via two different ways ; histogram and P-P secret plan. Figure 1 shows the histogram of arrested development standardized remainders. The histogram shows that there is a normal distribution of remainders. The frequence distribution of remainders is close to normal distribution line. Furthermore, figure 2 shows the P-P secret plan of arrested development standardized remainders and it shows that distribution of mistakes is normal. It can be said that first premise of multiple arrested development, normalcy, is non violated.
Figure 1 Histogram of Regression Standardized Residual
Figure 2 P-P Plot of Regression Standardized Residual
The 2nd premise of multiple arrested development to be checked is multicollinearity. Multicollinearity could be checked with correlativity matrix, VIF or tolerance values. There should non be any correlativity that is higher than.90 between two independent variables. When the correlativity matrix ( Table 3 ) is examined there is no correlativity higher than.90 between two independent variables. Table 4 shows the collinearity statistics of all three theoretical accounts. VIF values more than four or tolerance values higher than.20 are indexs of multicollinearity. Table 4 shows that there is no VIF value higher than four or tolerance value higher than.20. So, premise of multicollinearity is non violated.
Table 4
Collinearity Statisticss
Model
Collinearity Statisticss
Tolerance
VIF
1
( Constant )
Gender
.98
1.02
physical activity in a hebdomad
.98
1.02
2
( Constant )
Gender
.96
1.04
physical activity in a hebdomad
.94
1.06
milk cocoa V berry cocoa
.87
1.15
milk cocoa V peanut cocoa
.84
1.19
frequence of cocoa ingestion
.93
1.08
3
( Constant )
Gender
.92
1.08
physical activity in a hebdomad
.94
1.06
milk cocoa V berry cocoa
.86
1.17
milk cocoa V peanut cocoa
.80
1.24
frequence of cocoa ingestion
.84
1.19
fat rate ( % ) in cocoa
.67
1.49
chocolate tree rate ( % ) in cocoa
.70
1.43
The 3rd premise of multiple arrested development to be checked is homoscedasticity. Scatter secret plan of predicted value and residuary is used to command homoscedasticity. Any form should non be seen on the spread secret plan. Figure 4 shows that there is no form on the spread secret plan ; so, there is non homoscedasticity.
Figure 4 Scatter secret plan of predicted value and residuary
The 4th premise of multiple arrested development to be checked is independency. Independence is affected by the order of the independent variables and can be ignored if the order of independent variables is non of import. Order of the independent variables is of import in this survey ; so, independency should be checked in this survey. Independence is checked with Durbin-Watson value that should be between 1.5 and 2.5. Durbin-Watson value of the theoretical account is 1.88 ; so, independency premise is non violated.
The last premise of multiple arrested development is linearity. We assume that one-dimensionality is non violated in this survey.
Influential Observations
Datas should be checked whether there are outliers or non. Outliers could do deceptive consequences. There are different ways of look intoing outliers in multiple arrested development such as Partial secret plans, purchase statistics, Cook ‘s D, DFBeta and Mahalonobis distance. Each method uses a different computation method ; so, multiple methods should be used and so do a determination whether a information is outlier or non.
At first, partial secret plans of the dependant variable with each of the independent variable is examined ( see on figure 5,6,7,8 and 9 ) . Some instances that could be outliers are seen on each partial secret plan ; but, this should non be forgotten, doing determination over partial secret plans is a subjective manner and other ways of commanding outliers should be used. A determination could be made even after all methods were conducted.
Figure 5 Partial Plot of BMI and physical activity in a hebdomad
Figure 6 Partial Plot of BMI and milk cocoa V peanut cocoa
Figure 7 Partial Plot of BMI and frequence of cocoa ingestion
Figure 8 Partial Plot of BMI and fat rate in cocoa
Figure 9 Partial Plot of BMI and cacao rate in cocoa
After commanding partial secret plans, purchase value could be controlled to place the outliers. It is seen that there is no instance, purchase value of which is higher than.50. Harmonizing to leverage trial consequences there is no outlier.
Table 5
Extreme Values of Leverage Test
Case Number
Value
Centered Leverage Value
Highest
1
448
.04
2
384
.04
3
141
.03
4
324
.03
5
592
.03
Lowest
1
196
.00
2
103
.00
3
535
.05
4
160
.05
5
8
.05
After commanding purchase values, Cook ‘s distance could be controlled. In Cook ‘s Distance, a value greater than the value, calculated with the expression of average + 2 * standard divergence, can be admitted as outlier. In this survey critical value is.008 ( .002+2* ( .003 ) ) . Maximal value of Cook ‘s distance is.03 ; so, it is expected that there will be outliers. Boxplot of Cook ‘s distance ( figure 10 ) shows that the instances 499, 438, 449, 236, 284, 484, 37, 354, 137, 97, 324 and 165 could be outliers. On the other manus, harmonizing to Cook and Weisberg ( 1982 ) values greater than 1 could be admitted as outlier. So, it can be assumed that there is no outlier.
Figure 10 Boxplot of Cook ‘s distance
After commanding Cook ‘s Distance, DF Beta values of each independent variable could be checked. DF Beta value shows the alteration in arrested development coefficient due to omission of that row with outlier. Harmonizing to Field ( 2009 ) a instance can be outlier if absolute value of DF Beta is higher than one. Harmonizing to Stevens ( 2002 ) a instance can be outlier if absolute value of DF Beta is higher than two. In this survey there is no instance that has DF Beta value higher than one ( see figure 11 ) . Harmonizing to DF Beta trial values there is no outlier in this survey.
Figure 11 Boxplots of DF Beta values of Independent Variables
Last, Mahalanobis Distance could be controlled to place the outliers. If there is any instance that is greater than the value of qis square at I±=.001 that could be admitted as outlier. The critical value at I±=.001 with seven forecasters is 24.32. Table 6 shows the utmost values for this survey and there is no value greater than 24.32. Harmonizing to Mahalanobis distance trial there is no outlier.
Table 6
Extreme Values of Mahalanobis Distance
Case Number
Value
Mahalanobis Distance
Highest
1
448
23.72
2
384
20.90
3
141
20.50
4
324
19.15
5
592
17.99
Lowest
1
196
2.62
2
103
2.62
3
535
2.78
4
160
2.78
5
8
2.78
If the consequences of each trial is summarized ;
Partial secret plans shows that there could be outliers,
Leverage values show that there is no outliers,
Cook ‘s distance values show that there is no outlier,
DF Beta values show that there is no outlier.
Harmonizing to consequences of the trials, it could be assumed that there is no outlier.
Arrested development Consequences
A hierarchal arrested development analysis was conducted to place the forecasters of BMI. Three different theoretical accounts were examined to understand which forecaster explains has how much discrepancy. Table 7 shows the sum-up of three theoretical accounts. Among three theoretical accounts, the first theoretical account is non statistically important ; the 2nd and 3rd theoretical accounts are important.
In the first theoretical account ; gender and physical activity in a hebdomad were the forecasters. This theoretical account explains the.2 % of entire discrepancy, but insignificant ; F ( 2, 597 ) = .67 ; p & gt ; .05.
In the 2nd theoretical account, milk cocoa V berry cocoa, milk cocoa V peanut cocoa and frequence of cocoa ingestion are the forecasters after commanding for the consequence of gender and physical activity in a hebdomad. This theoretical account explains 13 % of entire discrepancy explained significantly, F ( 3, 594 ) = 28.901 ; P & lt ; .01.
In the 3rd theoretical account, cacao rate ( % ) in cocoa, fat rate ( % ) in cocoa are the forecasters of BMI after commanding for the consequence of gender, physical activity in a hebdomad, milk cocoa V berry cocoa, milk cocoa V peanut cocoa and frequence of cocoa ingestion. This theoretical account explains 34 % of entire discrepancy explained significantly, F ( 2, 592 ) = 189.154, P & lt ; .01.
Table 7
Arrested development Analysis Model Summary
Model
Roentgen
R2
Change Statisticss
Durbin-Watson
I”R2
I”F
df1
df2
I” Sig. F
1
.05a
.00
.00
.69
2
597
.50
2
.36b
.13
.13
28.90
3
594
.00
3
.69c
.47
.34
189.15
2
592
.00
1.879
a. Forecasters: ( Constant ) , physical activity in a hebdomad, gender
b. Forecasters: ( Constant ) , physical activity in a hebdomad, gender, milk cocoa V berry cocoa, frequence of cocoa ingestion, milk cocoa V peanut cocoa
c. Forecasters: ( Constant ) , physical activity in a hebdomad, gender, milk cocoa V berry cocoa, frequence of cocoa ingestion, milk cocoa V peanut cocoa, chocolate tree rate ( % ) in cocoa, fat rate ( % ) in cocoa
d. Dependent Variable: organic structure mass index
Table 8 shows the Coefficients of Hierarchical Regression Analysis that shows the significance and entire discrepancy explained by each forecaster. In the first theoretical account any of the forecasters significantly predicts the dependant variable, BMI. It can be said that neither the theoretical account, nor the forecasters are statistically important and do non foretell the result variable, F ( 2, 597 ) = .67 ; p & gt ; .05.
In the 2nd theoretical account, overall theoretical account is important, F ( 3, 594 ) = 28.901 ; P & lt ; .01 ) . In this theoretical account, merely one forecaster, milk cocoa V berry cocoa, is non statistically important. As milk cocoa is the mention class and milk cocoa V berry cocoa was found to be non important that means there is no important difference between milk cocoa and berry cocoa. As milk cocoa as the mention class and milk cocoa V peanut cocoa was found to be important that means there is a important difference between milk cocoa and peanut cocoa degrees. As positive relationship was found insignificant cocoa ‘s mean is higher than milk cocoa ‘s mean. Furthermore there is positive relationship between the frequence of cocoa ingestion and BMI. When 1 increases, the other will increase. Milk cocoa V peanut cocoa explains 3 % of entire discrepancy unambiguously. Frequency of cocoa ingestion explains 7 % of entire discrepancy explained.
In the 3rd theoretical account, overall theoretical account is important, F ( 2, 592 ) = 189.154, P & lt ; .01. In this theoretical account all forecasters are significantly predicts the dependant variable. There are positive relationships between fat rate in cocoa and BMI ; and, cacao rate in cocoa and BMI. Fat rate in cocoa explains 15 % of entire discrepancy unambiguously. Cacao rate in cocoa explains 4 % of entire discrepancy unambiguously.
Table 8
Coefficients of Hierarchical Regression Analysis
Model
Unstandardized Coefficients
Standardized Coefficients
T
P
Correlations
Bacillus
Std. Mistake
Beta
Part
1
( Constant )
24.419
.941
25.938
.000
Gender
-.232
.370
-.026
-.628
.530
-.026
physical activity in a hebdomad
.226
.251
.037
.900
.369
.037
2
( Constant )
17.165
1.309
13.110
.000
milk cocoa V berry cocoa
.539
.423
.052
1.273
.204
.049
milk cocoa V peanut cocoa
1.943
.420
.193
4.629
.000
.177
frequence of cocoa ingestion
1.751
.245
.283
7.135
.000
.273
3
( Constant )
5.426
1.191
4.557
.000
fat rate ( % ) in cocoa
.221
.017
.477
13.033
.000
.390
chocolate tree rate ( % ) in cocoa
.109
.016
.242
6.766
.000
.203
a. Dependent Variable: organic structure mass index
Discussion
Two different research inquiries were tried to be answered in this survey. First research inquiry was “ How good the type of cocoa and frequence of cocoa ingestion predict organic structure mass index, after commanding for gender physical activity? ” . Second research inquiry was “ How good do fat per centum and chocolate tree per centum in cocoa explain organic structure mass index, after commanding the consequences of the first research inquiry? “ .
A hierarchal arrested development analysis was conducted to reply the research inquiries. Three theoretical accounts were examined to happen the forecasters and their part to these theoretical accounts. The first theoretical account that examines that how good gender and physical activity in a hebdomad predict the dependant variable. Consequence of the first theoretical account shows that neither theoretical account nor forecasters significantly predict the BMI.
The 2nd theoretical account examined to reply the first research inquiry. This theoretical account predicts 13 % of entire discrepancy explained. Milk cocoa V berry cocoa does non significantly explicate the BMI. Milk cocoa V peanut cocoa explains 3 % , frequence of cocoa ingestion explains 7 % of entire discrepancy explained.
The 3rd theoretical account examined to reply the 2nd research inquiry. This theoretical account predicts 47 % of entire discrepancy explained and 34 % of entire discrepancy explained unambiguously. Fat rate in cocoa explains 15 % and chocolate tree rate in cocoa explains 4 % of entire discrepancy unambiguously.
When all theoretical accounts were examined it is seen that fat rate in cocoa is the best forecaster of BMI by explicating 15 % of entire discrepancy explained. Frequency of cocoa ingestion is the 2nd by explicating 7 % of entire discrepancy explained. Cacao rate is the 3rd forecaster by explicating 4 % of entire discrepancy explained.
