Role and Place of Statistical Data Analysis and very simple applications Simplified diagram of a scientific research When you know the system: Estimation of parameters and feedback When the system is too complicated

Knowledge New system Model Experiment Estimate Verify Predict A simple diagram of scientific research: When you know the system Data analysis

Simple application of statistics 1.Using previously accumulated knowledge you want to study a system 2.Build a model of the system that is based on the previous knowledge 3.Set up an experiment and collect data 4.Estimate the parameters of the model and change the model if needed 5.Verify if parameters are correct and they describe the current model 6.Predict the behaviour of the experiment and set up a new experiment. If prediction gives good results then you have done a good job. If not then you need to reconsider your model and do everything again 7.Once you have done and satisfied then your data as well as model become part of world knowledge Data Analysis is used at the stage of estimation and verification

Simple application of statistics The result of the model is usually a function dependent on two types of variables. The first type is that can be varied and the second type you want to estimate: Where x is a variable you can control and  is a variable you want to estimate. As a result of experiment you get observations for y. Then using one of the techniques (e.g. Maximum likelihood, Bayesian statistics) you carry out the estimation. Prediction is carried out for values of x that you have not done experiment for. In real life problem is more complicated. In many cases controllable and observations are dictated by the nature of experiment. But model is something different that is dependent on the parameters you estimate using this experiment I.e. experiment gives: But you want (you still need parameters):

Simple application of statistics You have a model and the results of experiment. Then you carry out estimation of parameters (e.g. using simplest least-squares technique): This simple estimation uses assumptions: Errors in experiment are independent, all of them have 0 mean and exactly same variance. After carrying out estimation of the parameters the next stage is to find out how accurate are they. Once this stage is complete, you carry out prediction (can you predict a value of y at the point x where you have not done experiment). If prediction at this stage works then model is fine. You give your results to a scientific community.

When system is too complicated Sometimes the system you are trying to study is too complicated to build a model for. For example in psychology, biology the system is very complicated and there are no unifying model. Nonetheless you would like to understand the system or its parts. Then you use observations and build some sort of model and then check it against the (new) data. Schematic diagram: Data (Design)Model Estimate Verify Predict Data analysis is used in all stages

When system is too complicated Usually you start from the simplest models: linear models. If linear model does not fit then start complicating it. By linearity we mean linear on the parameters. This way of modeling could be good if you do not know anything and you want to build a model to understand the system.

When system is too complicated In many cases just simple linear model may not be sufficient. You need to analyse the data before you can build any sort of model. In these cases you want to find some sort of structure in the data. If you can find a structure in the data then it is very good idea to look at the subject where these data came from. We will learn some of the techniques that can give some idea about the structure of the data. Usual techniques include: Principal component analysis, correspondence analysis, factor analysis, discriminant analysis, metric scaling, clustering.

When system is too complicated When system is too complicated, instead of building the model that can answer to your all question you sometimes want to know answer to simple questions. E.g. if effect of two or more factors are significantly different. For example you may want to compare the effects of two different drugs or effects of two different treatments. We will have a lecture about ANOVA and how to analyse the results using R. ANOVA is useful when you want to compare the effects of more than two factors.

When system complicated: Various criteria Occam’s razor: “entities should not be multiplied beyond necessity” or “All things being equal, the simplest solution tends to be the right one” A potential problem: There might be conflict between simplicity and accuracy. You can build tree of models that would have different degree of simplicity at different levels Rashomon: Multiple choices of models When simplifying a model you may come up up with different simplifications that have similar prediction errors. In these cases, techniques like bagging (bootstrap aggregation) may be helpful

Some application of data analysis Simplest application of statistics is: You have a vector of observations and you want to know if the mean is equal to some pre-specified value (say zero). Then you calculate mean value and check against this value. It is done by simple t-test. t.test(data) This command will calculate for you the mean, variance of the data and then calculate the relevant statistics. It will also give you confidence intervals. If the confidence interval does not contain the value you want to test against (say zero) then you can say that according to these data with 95% confidence that mean is not equal to zero. More over if p value is very small then you can say with 100-p*100 percent confidence that the value is different from zero

Some application of data analysis Another very simple application of statistics is comparing means of two samples using t.test. Before doing this test it is a good idea to have a look a box plot and test if variances are equal var.test(data1,data2) If it can be assumed that variances are equal then you can use t.test(data1,data2,var.equal=1) If variances are not equal then use t.test(data1,data2,var.equal=1)

Some application of data analysis If you can influence the experiment then you should emphasise the importance of paired designs. If design is paired then many systematic differences due to some unknown factors may be avoided. It is done easily using t.test again t.test(data1,data2,paired=1)