Last time out I looked at one of the complications of time series modelling with smoothers; you have a non-linear trend which may be statistically significant but it may not be increasing or decreasing everywhere.
To get a headstart on this, I’ll reuse the model we fitted to the CET time series from the previous post. Here, I’ll use a version of the Deriv() function used in the last post modified to do the posterior simulation; derivSimulCI(). In fitting the model we get a point estimate for each term, plus their standard errors in the form of the variance-covariance (VCOV) matrix of the terms2. Computers are good at simulating data and you’ll most likely be familiar with rnorm() to generate random, normally distributed values from a distribution with mean 0 and unit standard deviation. Returning to the simple linear regression case, let’s do a little simulation from a known model and look at the multivariate normal distribution of the model parameters.
The mean vector for the multivariate normal is just the set of model coefficients for mod, which are extracted using the coef() function, and the vcov() function is used to extract the VCOV of the fitted model. Each row of sim contains a pair of values, one intercept and one ( _x ), from the implied multivariate normal. 5000 random draws from the posterior distribution of the parameters of the fitted additive model. The large spread in the points (from top left to bottom right) is illustrative of greater uncertainty in the intercept term than in ( _x ). As I said earlier, each point on the plot represents a valid model consistent with the estimates we achieved for the sample of data used to fit the model.
The grey lines show the model fits for a random sample of 50 pairs of coefficients from the set of simulated values. You’ll be pleased to know that there is very little difference (non really) between what I just went through above for a simple linear regression and what is required to simulate from the posterior distribution of a GAM. Before we get to posterior simulations for the derivatives of the CET additive model fitted earlier, let’s look at some simulations for the trend term in that model, m2. Next, generate a small sample from the posterior of the model, just for the purposes of illustration; we’ll generate far larger samples later when we estimate a confidence interval on the derivatives of the trend spline.
The linear predictor matrix, lp, has a column for every basis function, plus the constant term, in the model, but because the model is additive we can ignore the columns relating to the nMonth spline and the constant term and just work with the coefficients and columns of lp that pertain to the trend spline.
As we saw in the previous post, the linear predictor matrix can be used to generate finite differences-based estimates of the derivatives of a spline in a GAM fitted by mgcv.
As I haven’t yet written a confint() method, we’ll need to compute the confidence interval by hand, which is no bad thing of course!
CI is now a list with two components, each of which contains a matrix with two rows (the two probability quantiles we asked for) and 200 columns (the number of locations at which the first derivative was evaluated).
First derivative of the seasonal and trend splines from the CET time series additive model. To leave a comment for the author, please follow the link and comment on their blog: From the Bottom of the Heap - R. The Greenlandic population, a small and historically isolated founder population comprising about 57,000 inhabitants, has experienced a dramatic increase in type 2 diabetes (T2D) prevalence during the past 25 years1.
Figure 2: Associations between 2-h plasma glucose levels and genotypes, as determined by Metabochip assay. Tests were performed using an additive linear mixed model in 2,540 individuals from the IHIT study who were not known to have T2D and for whom valid 2-h plasma glucose data were available.
The 1000 Genomes Project Consortium A map of human genome variation from population-scale sequencing.
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. I'm having troubles with understanding a problem which is basically asking to calculate a sample size (in order to estimate what portion of population has income more than 40K) with 95% confidence level and margin of error (E) of 1%.
Two things confuse me here: how to find maximum standard deviation and how quantile is different from a z-score?
Since this problem is routine bookwork, as one might do in a homework exercise, you should add the self-study tag and read its tag wiki info.
Your assertion that the correct value of the quantile should be 1.96 (if we assume the normal approximation is accurate) is completely correct.

However, the suggested value of 2 is a common approximation; it's only 2% larger, and that small additional margin by rounding the value up may be a good idea since several approximations are involved in doing the calculations. Secondly, if I told you the population proportion, could you compute the standard deviation? If you knew the population proportion and the sample size, could you compute the standard deviation?
Not the answer you're looking for?Browse other questions tagged self-study normal-distribution standard-deviation sample-size or ask your own question. Given a mean and 95% confidence interval, do I need to know the sample size to calculate the standard deviation? Instead of being a linear model in the original data however, the linear model is fitted using the basis functions as the covariates1. Taken together, the point estimates of the model terms and the VCOV describe a multivariate normal distribution.
Well, simulating from a multivariate normal is just as simple3, as long as you have the mean vector and the variance covariance matrix of the parameters. Several implementations are available in a range of packages, but here I’ll use the one in the MASS package which ships with all versions of R. However, instead of dealing with two or just a few regression coefficients, we now have to concern ourselves with the potentially larger number of coefficients corresponding to the basis functions that combine to form the fitted splines. If you look back at an earlier code block, I created a grid of 200 points over the range of the data which we’ll use to evaluate properties of the fitted model. And as we just went through, we can combine posterior simulations with the linear predictor matrix. We do this by by taking two extreme quantiles of the distribution of the 10,000 posterior simulations we generated for the first derivative at each of the 200 points we wanted to evaluate the derivative. If you allow gam() to select the degree of smoothness then you need to fit a penalized regression.
Motivated by this, we performed association mapping of T2D-related quantitative traits in up to 2,575 Greenlandic individuals without known diabetes. 2: Association between fasting plasma glucose level, fasting serum insulin level and 2-h serum insulin level and the SNPs on the Metabochip for the IHIT cohort.
4: Mean 2-h plasma glucose levels in the IHIT cohort stratified by Inuit admixture proportion. For each SNP, the ancestral state is coloured red, while the derived allele is coloured yellow. The standard deviation is not given and it says that I should take a maximum possible value for that. Even if you're not able to do the algebraic manipulation to solve for $n$, you can find this by trial and error.
The entire point is to calculate the sample size you need in that formula to get the desired margin of error.
In that post I explained how we can use the first derivatives of the model splines for this purpose, and used the method of finite differences to estimate them.
As with any other linear model, we get back from it the point estimate, the ( _j ), and their standard errors.
The only practical difference is that instead of multiplying each simulation by the observed data4 with mgcv we generate the linear predictor matrix for the observations and multiply that by the model coefficients to get simulations. Note that the code in derivSimulCI() is slightly different to this, but it does the same thing. One of the reasons I did 10,000 simulations is that for a 95% confidence interval we only need sort the simulated derivatives in ascending order and extract the 250th and the 9750th of these ordered values.
Sections of the spline where the confidence interval does not include zero are indicated by coloured sections. Superimposed are the first derivatives of the splines for 20 randomly selected posterior simulations from the fitted spline.
Plus, the time series models fitted to the CET data aren’t fitted via gam() but via gamm(), where we are using the observation that a penalized regression can be expressed as a linear mixed model, with random effects being used to represent some of the penalty terms.
Using array-based genotyping and exome sequencing, we discovered a nonsense p.Arg684Ter variant (in which arginine is replaced by a termination codon) in the gene TBC1D4 with an allele frequency of 17%.

Haplotypes are ordered according to the coding SNP rs61736969 (p.Arg684Ter), whose position is highlighted. Inuit health in Greenland: a population survey of life style and disease in Greenland and among Inuit living in Denmark. Postchallenge glucose, A1C, and fasting glucose as predictors of type 2 diabetes and cardiovascular disease: a 10-year prospective cohort study.
In addition, it says that 2 should be used as the quantile of the normal distribution of 0.975. To assess statistical significance of the derivative (the rate of change) I relied upon asymptotic normality and the usual pointwise confidence interval. Note that the point estimates are known as the mean vector of the multivariate normal; each point estimate is the mean, or expectation, of a single random normal variable whose variance is given by the standard error of the point estimate.
The estimates are both for the 2,733 individuals in the Greenlandic sample (IHIT), depicted to the left of the vertical line, and for 50 Danes, to the right of the vertical line. Large-scale association analyses identify new loci influencing glycemic traits and provide insight into the underlying biological pathways. Large-scale association analysis provides insights into the genetic architecture and pathophysiology of type 2 diabetes. A genome-wide approach accounting for body mass index identifies genetic variants influencing fasting glycemic traits and insulin resistance. The Metabochip, a custom genotyping array for genetic studies of metabolic, cardiovascular, and anthropometric traits. That interval is fine if looking at just one point on the spline (not of much practical use), but when considering more points at once we have a multiple comparisons issue. The colour of the circle indicates the extent of correlation (r2) between the SNP and p.Arg684Ter.
The estimates were achieved using the software tool RelateAdmix, which provides maximum likelihood estimates that take admixture into account by allowing alleles to be IBD only if the alleles are from the same population. The red dashed horizontal line indicates the 0.05 significance threshold after Bonferroni correction for multiple testing. Instead, a simultaneous interval is required, and for that we need to revisit a technique I blogged about a few years ago; posterior simulation from the fitted GAM.
The dashed horizontal line indicates a 0.05 significance threshold after Bonferroni correction for multiple testing. The circles representing p.Arg684Ter and rs7330796 are labelled (to the left of the circles). The circle representing rs7330796 has a label to the left of it so that it can be identified.
DIND is a measure of the ratio of the nucleotide diversity of haplotypes carrying the derived allele to the nucleotide diversity of haplotypes carrying the ancestral allele.
Analyses of skeletal muscle biopsies showed lower messenger RNA and protein levels of the long isoform of TBC1D4, and lower muscle protein levels of the glucose transporter GLUT4, with increasing number of p.Arg684Ter alleles. The solid blue curve (right y axis) illustrates the recombination rates from the Chinese HapMap (CHB) panel.
The DIND value for the coding SNP rs61736969 is highlighted by a dashed line and lies in the top 5% of the distribution.
These findings are concomitant with a severely decreased insulin-stimulated glucose uptake in muscle, leading to postprandial hyperglycaemia, impaired glucose tolerance and T2D.
The solid blue line illustrates the recombination rate from the Chinese HapMap (CHB) panel (in centimorgan (cM) per Mb, right y axis). The observed effect sizes are several times larger than any previous findings in large-scale genome-wide association studies of these traits2, 3, 4 and constitute further proof of the value of conducting genetic association studies outside the traditional setting of large homogeneous populations.
Exons are depicted as boxes, and the location of the p.Arg684Ter polymorphism is indicated by a red arrow.

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