Question

1. Using a short paragraph explain each part in 10 lines:

a) The meaning of an “interaction” term in a general linear model.

b) The importance of ensuring data points are independent.

c) The value of an orthogonal design.

d) The difference between a random and a fixed effect.

e) The difference between a general linear model and a generalized linear model

f) The difference between a Type I and Type II error.

Answer 1

a) The meaning of an “interaction” term in a general linear model.

An interaction term is that term which describes interactions or associations between two or more than two variables. An interaction occurs when an independent variable has a different effect on the outcome depending on the values of another independent variables , i.e the independent variables might interact with each other.

For example , we are working on a data about weight loss programe

let y denoted weigth of an indivisual which is dependent variable, and let boiled packed food and diet pill be two independent variables which are used in weight loss programe.

Now weigth may depend on only eating boiled packed food or taking diet pills only , or both independent variables i.e on interaction term of boiled packed food and diet pill . We can say that when boiled packed food is used with different dose or levels of diet pill helps in weigth reducing .

So we may add interaction term of boiled packed food and diet pill to our general linear model to get more information .

b) The importance of ensuring data points are independent.

Two observations or data points are said to be independent if the occurrence of one observation does not depends on previous observations , or provides no information about the occurrence of the other observation.

The assumption of independence of observations or dat points is used for t-tests, ANOVA tests, and in several other statistical tests.

For example suppose we want to test two drugs which claims to reduce headache . So we want new observation ( here may be animals or human ) on whom the drugs will be tested . So we make two gropus and 1st drug is given to first group and 2nd drug to other group.

As effects of drug may depend on person age, genetics , weight ect.

Thus we also need to assign every person randomly to each gropus .Thus independence of observation is very important as if peoples of same family or friends of same age ( which are dependent on each other) are put into one group , result of drug will not be satisfing or it may give wrong results.

Also we don’t want one person appearing twice in two different groups as it could skew your results.

c) The value of an orthogonal design.

Orthogonal Design (Experimental Design) -An experimental design is orthogonal if each factor can be evaluated independently of all the other factors.

An orthogonal model means that all independent variables in that model are uncorrelated. If one or more independent variables are correlated, then that model is non-orthogonal.

Orthogonality refers to the property of a design that ensures that all specified parameters may be estimated independent of any other. The degree of orthogonality is measured by the normalized value of the determinant of the information matrix.

An orthogonal design matrix having one row to estimate each parameter (mean, factors, and interactions) has a measure of 1. It is easy to check for orthogonality: If the sum of the factors columns in standard format equals 0, then the design is orthogonal.

d) The difference between a random and a fixed effect.

A Random effects model is a statistical model where the model parameters are random variables , like the price for a cup of cofee varies wildly depending on location and brand .

Where as in fixed effect variables are constant across individuals; these variables, like age, sex, or ethnicity, don’t change or change at a constant rate over time .

They have fixed effects,in other words, any change they cause to an individual is the same. where as The Random effects model is a special case of the fixed effects model.

Fixed effects do have some limitations for example, they can’t control for variables that vary over time which we do not observe in case of Random effects i.e Fixed effects are constant across individuals, and random effects vary

Fixed effects model refers to a regression model in which the group means are fixed (non-random) as opposed to a Random effects model in which the group means are a random sample from a population.

e) The difference between a general linear model and a generalized linear model

The General Linear Model (GLM) is a useful statistical method for comparing how several variables affect different continuous variables ( dependent variable aftected by other independent variables ).

General Linear Model is the foundation for several statistical tests, including ANOVA, ANCOVA and regression analysis. Despite their differences, each fits the definition of Data = Model + Error , we we assume that Error follows normal distribution.

Where as the generalized linear model (GLM) is a flexible generalization of ordinary linear regression that allows for response variables that have error distribution models other than a normal distribution.

Thus , general linear model requires that the response variable follows the normal distribution while the generalized linear model is an extension of the general linear model that allows the specification of models whose response variable follows different distributions .

f) The difference between a Type I and Type II error.

Type I error is Probability of rejection of a null hypothesis, when infact it is true

A type 1 error is also known as a false positive and occurs when a researcher incorrectly rejects a true null hypothesis.  

i.e Type I error = P(Reject H0 | H0 is true )

Where as Type II error is Probability of accept the a null hypothesis, when infact it is false

A type II error is also known as a false negative and occurs when a researcher fails to reject a null hypothesis which is really false.

i.e Type II = P(Accept H0 | H1 is true ) = P ( Reject H1 | H1 is true )

The probability of making a type I error is represented by your alpha level , whereas The probability of making a type II error is called Beta