Statistical genetics: Chapter 2, the concept of statistical analysis

Hello everyone , I'm brother Fei .

I recommended this book a few days ago , Can receive pdf And supporting data code . here , I will introduce each chapter , Summary is also a learning process .

The quotation part is the Google translation of the original book , The body part is my understanding .

Part I foundation , It is divided into six chapters , Namely ：

• Chapter one ： Basic concepts of genome （ This section introduces , Click to enter ）
• Chapter two ： The concept of statistical analysis
• The third chapter ： Genotype data parameters
• Chapter four ：GWAS analysis
• The fifth chapter ： Polygenic effect
• Chapter six ： Genes interact with the environment

today , Introduce the contents of Chapter 2 , The concept of statistical analysis , Take a look at the catalog ：

primary coverage

This chapter includes ：

• Basic statistical concepts , Including variance 、 Average , Standard deviation , covariance , And the variance covariance matrix
• The basic framework of statistical model , Include ： Invalid Hypothesis 、 The alternative hypothesis , Significance threshold
• Correlation and causality
• Causal model
• Fixed effect model 、 Random effect model and mixed effect model
• I've understood fitting

Why study statistics

up to now , We have been focusing on mastering the basic concepts and foundations of the human genome . Before moving on to a more advanced topic , Especially before the applied statistics chapter later in this book , You must also master some core statistical concepts . As we pointed out at the beginning , This book is written at an introductory level , It aims to cater to various groups of researchers who first entered this field . Readers who already have some basic statistical knowledge will feel that this chapter is too introductory . It's been a long time for those statistics courses , Or just people who have studied statistics as part of a larger course , You may find this basic review section useful . Although many of the concepts introduced in this chapter are familiar in non genetic data analysis , But we also emphasized the special statistical concepts and problems of genetic data analysis .

The purpose of this chapter is to provide basic introductory knowledge for understanding the core concepts required for genetic data analysis . then , We will cover more advanced topics that you may wish to explore further . If you are interested in applying these techniques at a higher level of statistical complexity , We strongly encourage readers to use more advanced or professional textbooks for further reading , Some of them are mentioned in the further reading section at the end of this chapter . We first review some basic statistical concepts that are not unique to statistical genetic data analysis , Such as the central trend . then , Most of the content in this chapter is about the basics of statistical models , And provides updates to related concepts , Such as null hypothesis, substitution hypothesis and significance threshold . then , We distinguish between the concepts of correlation and causality , This is critical when evaluating these models . We also guide the reader through various types of causal models , Including direct and two-way causality 、 Additive and common cause models , And joint mediation and detente （ Or interaction ） Model . Next, the commonly used fixed effects are briefly introduced 、 Difference between random effects and mixed models . Last , We are interested in copying 、 Overfitting is briefly discussed , Then a brief summary is provided .

Flying brother's notes ： Statistical genetics , It's not just genetics 、 Molecular biology , We should also learn statistics , Describe some characteristics of genetics through statistical parameters , Summarize some rules , It's very important .

Basic statistical concepts

Because we know that readers may come from different disciplines and backgrounds , So let's start by introducing the main concepts . As the first 1 Chapter , You often encounter the term phenotype （ In these statistical models are dependent variables ） And genotype （ It is often called a covariate 、 Predictors or independent variables ）. As we will explain in the following chapters , These variables can take different forms according to their measured values . This measurement in turn affects the statistical model we choose . for example , If a phenotype is measured as a binary variable （1= disease ,0= No disease ）, Then you will use logic or other similar models . However , If the phenotype is taken as a continuous or quantitative result （ Height, for example ） Take measurements , You need to use a model , The model can not only capture the gradual scale of the data , And it is usually possible to capture the distribution of this measurement .

Flying brother's notes ： In statistical genetics , There is very little analysis of variance , Are regression analysis , The two taxonomic characters are logistic Model , Continuous traits are general linear models or mixed linear models . Heritability 、 Genetic correlation is calculated from variance components ,snp Significance is the significance test of regression , Polygenic scores are predictive regression models, and so on

Average 、 Standard deviation and variance

These parameters , It generally refers to the continuous character of normal distribution ：

The formula of sample variance ：

R Code display ：

Simulate a data frame ,20 Data ：

library(tidyverse)
dat = data.frame(ID = 1:20, y = rnorm(20)+100)
dat

Calculate average ：

> mean1 = mean(dat$y);mean1
[1] 100.2073

Calculate variance ：

> var1 = var(dat$y);var1
[1] 1.27476

Calculate the standard deviation ：

> sd1 = sd(dat$y);sd1
[1] 1.129053

Another way to calculate variance ：

> sum((dat$y - mean1)^2)/(20-1)
[1] 1.27476

Variance covariance matrix

X and Y The covariance of is expressed as COV(X,Y),X and X The covariance of is X The variance of .

If there are multiple variables , Used to represent their variance covariance matrix , It can be written like this , The diagonal is the variance , The non diagonal is the covariance between two pairs .

R Code demonstration ：

y1 and y1 The covariance , Consistent with variance ：

> cov(dat$y1,dat$y1)
[1] 0.9460778
> var1 = var(dat$y1);var1
[1] 0.9460778

y1,y2,y3 Variance covariance matrix of , The diagonal is the variance , The non diagonal is the covariance between two pairs .

> cov(dat[,2:4])
            y1          y2         y3
y1  0.94607782 -0.07404345 -0.1165461
y2 -0.07404345  0.68879820  0.0776964
y3 -0.11654608  0.07769640  0.9165009
> cov(dat$y1,dat$y2)
[1] -0.07404345
> cov(dat$y1,dat$y3)
[1] -0.1165461

Statistical models

The regression model

Polygenic score ：Polygenic score

If y = a*x + b,a Is the regression coefficient ,b It's intercept .

Calculation method of regression coefficient ：

Be careful , The denominator here is x The variance of .

R Code , You can see two ways , The calculation results are consistent ：

> lm(dat$y2 ~ dat$y1)

Call:
lm(formula = dat$y2 ~ dat$y1)

Coefficients:
(Intercept)       dat$y1  
  207.78619     -0.07826  

> cov(dat$y2,dat$y1)/var(dat$y1)
[1] -0.0782636

Null hypothesis and significance test

The goal of regression methods is usually to test the null hypothesis , This is a statistical test , Used to determine that there are no significant differences between specific groups . Think back to your previous introductory course in Statistics , This refers to your estimated parameters （β,β） Equal to zero . therefore , Another assumption is that when the parameter is not equal to zero . We use data for statistical tests , If the null hypothesis is true , The calculation p Value to determine statistical significance . In short , If p Small values , Then the data is inconsistent with the null hypothesis . If the parameter passes the significance threshold （ for example ,0.05,0.001）, The null hypothesis will be rejected , To support alternative assumptions . In the area of statistical significance , There was considerable criticism and heated discussion , Mainly around the fact that , That is, the results are often only related to invalid assumptions .

relevant 、 Causal and multiple causal models

In this book , relevant （r） The terms cause and effect are frequently used , Therefore, it is necessary to distinguish the two terms . Correlation represents the statistical correlation between two variables . It is a scaled version of covariance , Its value is between -1 and 1 Between . therefore , A value close to zero means that the covariance between variables is very small . near 1 The value of represents a strong positive covariance , Close to negative 1 The value of represents a strong negative covariance . Please note that , The covariance we are discussing here 、 Correlation and regression models assume that there is a linear relationship between the two variables . In other words , We will draw a line through the scatter , And assume that one variable in its distribution is based on a constant unit change of another variable . When the correlation is close 1 when , Variables tend to move in the same direction , The change around the association line is relatively small ; When approaching -1 when , In the opposite direction .

The formula of correlation coefficient ：

stay R In language , Yes cor function , The correlation coefficient can be calculated directly , The correlation coefficient can also be calculated by the above formula , Let's compare the two through code ：

> cor(dat$y1,dat$y2)
[1] -0.09172278
> 
> cov(dat$y1,dat$y2)/(sd(dat$y1)*sd(dat$y2))
[1] -0.09172278

Causality and correlation

Correlation , Not necessarily cause and effect , such as ：

Correlation simply describes the size and direction of the relationship between two or more variables . This does not mean that a change in one variable causes a change in the value of another variable . contrary , Causality indicates that , A change in the value of one variable is the result of a change in another variable . This is called causality .

Take smoking for example . You will find that smoking may be highly correlated with other behaviors , For example, high alcohol consumption . However , You can't naively infer that smoking can lead to alcoholism . Causality is often difficult to reveal in almost all types of data analysis , Including statistical genetic data analysis .

How to determine the causal relationship ？

In medical research , Using randomized controlled trials is the most effective way to establish causality , In these types of research , Samples are usually divided into two groups , These two groups are in most WAV Chinese are similar . Then they receive different treatments , For example, a placebo with a specific treatment or drug . Then evaluate the results of each group . If the results are significantly different . Cause and effect can be established .

in fact , Many of the complex features we study in this field tend to avoid case-control or randomized control methods , We can observe environmental changes （ for example , Policy changes 、 Contamination exposure ）, But determining causality remains a challenge .

In the later chapters of this book , We will show a variety of applied statistical methods , These methods attempt to establish causal relationships in statistical genetics , Including Mendelian randomization . We will also highlight the challenges and areas of ongoing debate in this study . Most applications of quantitative statistical genetics are aimed at estimating multivariate statistical models , This is the topic we are going to discuss now .

Fixed model 、 Random model and mixed linear model

Now we only discuss the fixed effect model , That is, the effect of covariates on phenotypic outcomes is modeled as fixed or the same increase per unit of covariates in the sample . therefore , Fixed effect model is different from random effect model or mixed model , Some or all of the model parameters are considered as random variables . The reader should pay attention to , These terms are used slightly differently in biostatistics and econometrics . Andrew · Gelman （AndrewGelman） Wrote an excellent blog , Describe these differences . In Econometrics , Fixed effect models are often used to determine a specific set of variables contained in hierarchical or panel data . In the text of Biostatistics and genetics ,“ Fixed effect ” It refers to the average population effect “ Random effects note the distribution of subitem specific effects . These random subject specific effects are usually considered as unknown potential variables .？2 We often use these models to control so-called unobserved heterogeneity . The assumption here is usually that heterogeneity is a time constant , Independent of other covariates . Random effect models are often very useful because we have a subset of individuals in the data . This includes changes in subsets of individuals or clusters , Such as family 、 School 、 Community 、 City 、 Country or hospital . When checking longitudinal data , Subsets can be repeated measurements of individuals . perhaps , If you check the recurrence event data , Subsets may be repeated disease episodes . therefore , We model random effects , To explain the subset of the data that may in turn affect the main effect .

The mixed linear model contains fixed and random effects . They are often used to examine repeated measurements of the same individual or a specific subset of measurements in longitudinal group studies . In the genetic research covered in this book , Hybrid models are useful for controlling population structure and estimating heritability .

When these models are applied to group structure , Random effect is the effect of genotype on the correlation between individuals - Contribution of phenotypic association . As mentioned earlier , Correlations between individuals are made using the genome relationship matrix （GRM） Calculated . therefore , The hybrid model can explain the genetic distance between individuals in the sample , So as to control the potential confusion caused by the correlation between genetic map differences and geographical differences .

As we discussed in the heritability section of Chapter 1 , Hybrid models are also commonly used to use genome-based restricted maximum likelihood （GREML） It is estimated that SNP Heritability . This is a statistical method of variance component estimation , Narrow additive contribution used to quantify phenotypic heritability . This is specific to a particular subset of genetic variation , It's usually limited to MAF Greater than 1% The locus of . For this reason , It's often called “ chip ” or “ Single nucleotide polymorphism ” Heritability （ or h2SNP, As the first 1 Chapter ）. As mentioned earlier , With the advent of genome-wide data . Researchers can go beyond using twin models to test genetic similarity between unrelated individuals . The software used for this analysis is GCTA- Genome wide analysis of complex traits （ See at the end of this chapter “ Mixed model analysis software ”）. These estimates produce a lower limit on the genetic contribution of a phenotype or trait , Without relying on the often limited assumptions in twins or family analysis . In short , If a particular phenotype is heritable , Then individuals with closer genetic relationships should have more similar phenotypic values . If the genetic correlation of individuals is not an indicator of similar phenotypic values , So we can come to a conclusion , Specific phenotypes may not be affected by genetics . At the end of this book 9 In the chapter , We provide a how to use GCTA And an example of such analysis .

Flying brother's notes ： Mixed linear models are often used in animal and plant breeding , In human statistical genetics , To estimate heritability, use GREML Methods to estimate variance components and calculate heritability , It uses genotype data （SNP） Built G The matrix is put into the random factors in the mixed linear model , Similar to... In genome selection GBLUP Method , Commonly used in humans GCTA To estimate , The two are equivalent .

Software for evaluating heritability ：

• GCTA：https://cnsgenomics.com/software/gcta/#Overview
• FastLMM：http://research.microsoft.com/en-us/um/redmond/projects/mscompbio/fastlmm/
• GEMMA：http://www.xzlab.org/software.html
• MMM：http://www.helsinki.fi/mjxpirin/download.html.

There are other soft estimates used in the summary of animals and plants ：

• R package ：asreml,sommer,BGLR
• ASReml,DMU,BLUPF90,HIBLUP etc.

The results were repeated and over fitted

If only one data set or sample is used for analysis ,“ You may encounter over fitting problems , This is related to the ability to replicate results in individual samples .

Over prediction refers to the problem that the prediction results of the predictors in the model in a specific data or sample are better than those in a new independent data set .

First , Over fitting may be the result of multiple tests . This is because our association between covariates and phenotypes violates such a basic premise , That is, this association is the result of real group effect and random chance . As we are in the 4 Described in Chapter , The genetic variation of these techniques is to test that sometimes there are millions of genetic variations with specific phenotypes . Organizations with the largest associations are more likely to make greater contributions than we expected , Not by chance . When researchers try to replicate the results in a smaller sample , They found that the result was often a small correlation . This is due to the fact that , That is, the top result of the initial over prediction model and its effect estimation （ For example, regression coefficient ） Greater or exaggerated the real effect . in fact , Any joint model with multiple covariates or predictors , If you build and test on a single sample , Will be over fitted .

This is because we estimate parameters to optimize the fit of the model to specific data . therefore , It is logical that the model does not perform well on new independent data .

In this introductory textbook , We can't describe all the ways to deal with over decoration , Only a few of them are outlined here . One is to use training and validation data sets , It is now more commonly used to solve this replication problem . One option is to retest the discovery in similar independent data sets , To see if the results repeat . Another option is to segment the data in the same sample into a training and verification set , This choice is due to the British biological bank （ Possession 50 10000 people ） Such as the release of large data sets . This can then be repeated with different data partitions , To improve robustness .

Other techniques for dealing with this problem are called regularization or contraction methods .

The shrink method performs variable selection , With valid shrink parameters , Keep the predicted value in the model , But it will shrink some parameter estimates . Lasso regression can be used to perform variable selection and ridge regression , To reduce the parameter estimation . Although they are far beyond the scope of this introductory article , But they effectively penalize the parameters in the formula for optimization . There is also an elastic net method combining ridge regression and lasso regression . In Bayesian contraction method , The penalty is expressed as a priori probability . Punishment can be set in many ways , For example, punish the big influence or reduce the small influence to zero or close to zero . The choice depends on the model and analysis . Because the basic facts are often unknown , Therefore, multiple analyses should be performed for testing , So as to make predictions in independent data and cross validation . These methods are becoming more and more common in genetics , Interested readers should refer to more advanced materials .

Flying brother's notes ： Build the model , Over fitting often occurs , For example, you dig in a group GWAS site , And then use significant in this group SNP forecast , The accuracy of the discovery is very high , But it is very low or ineffective in other groups , This is the typical performance of over fitting . The solution is to enlarge the sample size , Modeling in large samples , The strong type will be better , The other is to choose other algorithms , There are punishments in it , Such as ridge regression and LASSO Regression, etc