Simulation Study Clause Samples

Simulation Study. In order to study the performance of the cχ2 and normal distributions as approximations of the null distribution of the score statistic, we performed a simulation study. For sake of simplicity we used the data structure of our example of 33 families (see below). We generated 100,000 data sets of inde- pendently binomially distributed outcomes and 100,000 data sets of indepen- dently normally distributed outcomes. The score statistics were calculated using correlation structure (2.1) based on the coefficients of relationship. We also studied the performance of the distributions in a very small set of nine families. In table 2.1, the actual p-values corresponding to a nominal p-value of 0.05, 0.01, 0.001 and 0.0001 are given. The results were in favour of the cχ2 distri- bution for both binomially and normally distributed outcomes. Even for the set of nine families, the cχ2 distribution performed very well. TABLE 2.1: Type I error rate when using cχ2 distribution and normal distribution as approximation for the distribution of Q under the null hypothesis. The estimates are based on 100,000 simulations. 33 families 9 families nominal cχ2 normal cχ2 normal Binomial (DM2) 0.05 0.0547 0.0606 0.0550 0.0649 0.01 0.0143 0.0194 0.0137 0.0239 0.001 0.0020 0.0041 0.0017 0.0070 0.0001 0.0004 0.0011 0.0002 0.0019 Normal (BMI) 0.05 0.0538* 0.0615* 0.0566 0.0651 0.01 0.0125* 0.0196* 0.0151 0.0233 0.001 0.0016* 0.0047* 0.0027 0.0069 0.0001 0.0002* 0.0011* 0.0004 0.0023 To illustrate the score statistic, we used data from 79 patients with type 2 diabetes mellitus (DM2), their first-degree relatives and spouses (▇▇▇▇▇▇▇▇▇ et al., 2003). These families were derived from the GRIP population (Ge- netic Research in Isolated Populations), an isolated village in the Southwest of the Netherlands. The GRIP population is described in detail elsewhere (▇▇▇▇▇▇▇▇▇ et al., 2003; ▇▇▇▇▇▇▇ et al., 2002; ▇▇▇ ▇▇▇▇▇ et al., 2001). Probands are patients with DM2 treated by physicians participating in GRIP. Among the relatives are patients not related to ascertainment namely patients of other physicians and subjects who did not know that they have DM2. In a combined linkage and association study, a genome scan was carried out on these data and ▇▇▇▇▇▇▇▇▇ et al. (2003) found a borderline association between marker D3S3681 and DM2 (LOD score of 1.20, P=0.01). For DM2 we analysed 33 families informative for linkage. One of these families was a combination of two nuclear families. Three families had ...

Simulation Study. For each of the simulation scenarios, 5000 simulations were performed in R. Datasets from the first two simulation studies were simulated to resemble actual motivating data described in Section 3.2, with sample size N = 672. Independent predictor variables were generated to mimic age (years), smoking status (yes/no), race (1 = white / 2 = black), and SA status (yes/no), and the outcome variable was generated to resemble the cytokine MCP1 (µg/mL) based on a lognormal regression against those predictors. Age was simulated as a normal random variable with mean 26.6 and standard deviation 6.4, then rounded to the nearest whole number (this permits the formation of x-homogeneous pools when average pool size is small). Smoking status, race, and SA status were simulated as Bernoulli random variables with probabilities 0.47, 0.28, and 0.46, respectively. The outcome, MCP1, was generated under a lognormal distribution such that E[log(MCP1)|X] = —2.48+0.017(Age)+ 0.007(Smoking Status) — 0.388(Race) + 0.132(SA) and V ar[log(MCP1)|X] = 1.

Simulation Study. Acute antinociception paradigm Chronic analgesia paradigm

Simulation Study. In this Section we study the gain obtained in the prediction of direct estimators of the FGT poverty measures (for α = 0), in the sense of mean squared error of small area predictors, when considering three different approaches in the computation of matrix W in (4.5). The first one is the classical approach, where W is a typicality matrix, whereas the second and third approaches consist in implementing the techniques described in Sections 5.2 and 5.3, respectively. We start by describing the data to be used in the model described in Section 4.2. They consist of official data from the Spanish Survey of Income and Living Conditions corresponding to year 2006 for D = 51 Spanish provinces (the small areas). The response variable is the direct estimator of the FGT poverty measure (for α = 0), that is the proportion of poor in the area. The auxiliary covariates are the intercept and the following proportions (in the area) of Spanish people, people of ages from 16 to 24, from 25 to 49, from 50 to 64, equal or greater than 65, people with no studies up to primary studies, Graduate people, employees, unemployed people, inactive people. We have selected from the Instituto Nacional ▇▇ ▇▇▇▇▇´ıstica website (▇▇▇▇://▇▇▇.▇▇▇.▇▇), the more relevant socioeconomic variables related with poverty, being the unemployment rate and share of illiterate population over 16 years old. These variables have been measured in the D = 51 provinces from × 1991 to 2005 (J = 15 years). Therefore, in practice we have two matrices, X1 and X2 of size 51 15. × In order to compute matrix W with the multivariate approach of Section 5.2, we only consider the information contained in J-th columns of X1 and X2, which leads to a matrix of size 51 2. We call WM the proximity matrix computed with the methodology described in Section 5.2. To compute matrix W using the functional approach of Section 5.3, we have obtained two semi- metrics D(2) and D(2), one for each data set (see Ferraty and ▇▇▇▇ (2006)), for q = 4 functional principal 1,q 2,q components, since q = 4 is enough to collect the most part of the observed variability. Finally, in order to obtain a square matrix of joint distances from the previous two, we have used the related metric scaling technique, introduced by ▇▇▇▇▇▇▇ and Fortiana (1998), which provides a joint metric from different metrics on the same individuals, taking into account the possible redundant information that can be added simply by adding distance matrices. We call WF the...

Simulation Study. By means of a simulation study, we first evaluated the type I error rate of the score statistic Sˆ 1 , ▇▇▇▇▇▇▇’▇ chi square χ2, the likelihood ratio with equal weights LR, and the Terwilliger’s likelihood ratio with weights equal to pj’s TLR. For the score statistic we used the chi square distribution with one degree of freedom to approximate the distribution under the null hypothesis. For the LR and TLR statistics we used the 50:50 mixture of two chi squares with zero and one degree of freedom. We generated 10,000 samples of 200 case chromo- somes and 200 control chromosomes from the multinomial distributions with probabilities p1 pm for m equal to 4, 5, 8, 10, 15 and 20 haplotypes. Similar to the simulation described by ▇▇▇▇▇▇▇▇▇▇▇, ▇▇▇ frequency of the most common haplotype, p1, was set to 0.5, whereas the remaining haplotypes were equally frequent (0.5/(m — 1)). The results are shown in left columns of table 6.1. For all m, the type I error rates of the score statistic Sˆ 1 were maintained at the nominal error rate. For m < 10, the type I error rates of ▇▇▇▇▇▇▇’▇ chi

Simulation Study. The BioCycle Study, conducted from 2005 to 2007, followed premenopausal women from Western New York State for one or two complete menstrual cycles. Regularly menstruating women not currently taking oral contraceptives were eligible for participation. 259 women between the ages of 18 and 44 completed the study. Data collected during the study included age (years) and BMI (kg/m2), as well as serum estrodial, vitamin E, and HDL levels, which were measured on the 22nd day of a participant’s menstrual cycle. Participant BMI was right-skewed, with values ranging from 16.1 to 35.0, with an average body mass index of

Simulation Study. The aim of the simulation study is to evaluate empirically the power of the score test Sˆp in comparison with the Pearson’s χ2, Zˆmax and Zˆclump tests. We generated at least 1000 replicates from the multinomial distributions ac- cording to the models described previously. Without loss of generality we assumed that the first or first two haplotypes are associated with the disease. The remaining haplotypes were equally frequent. We varied the number of variants m from 3 to 20. The p-values of the test statistics were calculated em- pirically by means of ▇▇▇▇ ▇▇▇▇▇-▇▇▇▇▇ permutations using a program based on the program Clump (Sham and ▇▇▇▇▇▇, 1995). We used a nominal p-value of 0.05. To verify whether Monte-Carlo yields the right type I error rate of these test statistics, data sets were generated under the null model (λ = 0) each time for markers with 5, 7, 9, 11, 16 and 20 alleles. The frequency of the first allele was set to 0.5, whereas the remaining alleles were equally frequent. The results are shown in Table 4.1. The type I error rate is approximately equal to the nominal rate for the score Sˆp, ▇▇▇▇▇▇▇’▇ χ2, and Zˆclump tests, regardless of the number of alleles m at the marker locus, whereas the Zˆmax becomes somewhat conservative as the number of marker alleles m increases (Sham and ▇▇▇▇▇▇, 1995). TABLE 4.1: The type I error rates based on 10000 simulated m-by-2 tables for λ = 0 and p1 = 0.5. α m χ2 Zˆclump Zˆmax Sˆp m χ2 Zˆclump Zˆmax Sˆp 0.05 5 0.053 0.053 0.047 0.054 11 0.047 0.046 0.039 0.046 0.01 0.011 0.011 0.009 0.011 0.010 0.010 0.008 0.010 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.000 0.001 0.05 7 0.051 0.048 0.045 0.052 16 0.049 0.049 0.040 0.047 0.01 0.011 0.010 0.009 0.010 0.011 0.011 0.007 0.010 0.001 0.001 0.001 0.000 0.001 0.001 0.001 0.000 0.001 0.05 9 0.051 0.052 0.044 0.052 20 0.052 0.052 0.035 0.053 0.01 0.001 0.011 0.008 0.010 0.011 0.011 0.008 0.010 0.001 0.002 0.002 0.001 0.001 0.001 0.002 0.001 0.002 To study the power of the statistics we first considered the model used by Terwilliger (1995) for one positively associated common haplotype. The fre- quency p1 of this haplotype was 0.5 in controls. The parameter λ was fixed to 0.5, which corresponds to a haplotype frequency of 0.75 in the cases and a relative risk γ of 3. We considered 100 case chromosomes (n1) and 100 con- trol chromosomes (n2). The results are shown in Table 4.2. For m 5 all test statistics performed well; however Sˆp had slightly higher power than other t...

Simulation Study. I conducted four sets of real-data based simulation studies to demonstrate advantages of IPBT over existing methods.

Simulation Study. I conducted two sets of data based simulation studies to illustrate the consistency of gene panels and its applications to DE gene detection.