StatisticsChapter 1Why is my evil lecturer forcing me to learn statistics? Initial observation: finding something that needs explainingTo see whether an observation is true, you need to define one or more variables to measure that quantify the thing you’re trying to measure.Generating and testing theories and hypotheses A theory: an explanation or set of principles that is well substantiated by repeated testing and explains a broad phenomenon.A hypotheses: a proposed explanation for a fairly narrow phenomenon or set of observations.An informed, theory-driven attempt to explain what has been observed.A theory explains a wide set of phenomena with a small set of well-established principles.A hypotheses typically seeks to explain a narrower phenomenon and is, as yet, untested.Both theories and hypotheses exist in the conceptual domain, and you cannot observe them directly.To test a hypotheses, we need to operationalize our hypotheses in a way that enables us to collect and analyse data that have a bearing on the hypotheses.Predictions emerge from a hypotheses. A prediction tells us something about the hypotheses from which it derived.Falsification: the act of disproving a hypotheses or theory.Collecting data: measurement Independent and dependent variable Variables: things that can changeIndependent variable: a variable thought to be the cause of some effect.Dependent variable: a variable thought to be affected by changes in an independent variable.Predictor variable: a variable thought to predict an outcome variable. (independent)Outcome variable: a variable thought to change as a function of changes in...

StatisticsChapter 6The beast of bias Bias: the summary information is at odds with the objective truth.An unbiased estimator: one estimator that yields and expected value that is the same thing it is trying to estimate.We predict an outcome variable from a model described by one or ore predictor variables and parameters that tell us about the relationship between the predictor and the outcome variable.The model will not predict the outcome perfectly, so for each observation there is some amount of error.Statistical bias enters the statistical process in three ways:things that bias the parameter estimates (including effect sizes)things that bias standard errors and confidence intervalsthings that bias test statistics and p-values An outlier: a score very different from the rest of the data.Outliers have a dramatic effect on the sum of squared error.If the sum of squared errors is biased, the associated standard error, confidence interval and test statistic will be too. The second bias is ‘violation of assumptions’.An assumption: a condition that ensures that what you’re attempting to do works.If any of the assumptions are not true then the test statistic and p-value will be inaccurate and could lead us to the wrong conclusion.The main assumptions that we’ll look at are:additivity and linearitynormality of something or otherhomoscedasticity/ homogeneity of varianceindependenceAdditivity and...

StatisticsChapter 8Correlation The data we observe can be predicted from the model we choose to fit the data plus some error in prediction.Outcomei = (model) + erroriThusoutcomei = (b1Xi)+erroriz(outcome)i = b1z(Xi)+erroriz-scores are standardized scores.A detour into the murky world of covariance The simplest way to look at whether two variables are associated is to look whether they covary.If two variables are related, then changes in one variable should be met with similar changes in the other variable.Covariance (x,y) = Σni=1 ((xi-ẍ)(yi-ÿ))/N-1The equation for covariance is the same as the equation for variance, except that instead of squaring the deviances, we multiply them by the corresponding deviance of the second variable.A positive covariance indicates that as on variable deviates from the mean, the other variable deviates in the same direction.A negative covariance indicates that as one variable deviates from the mean, the other deviates from the mean in the opposite direction.The covariance depends upon the scales of measurement used: it is not a standardized measure.Standardization of the correlation coefficient To overcome the problem of dependence on the measurement scale, we need to convert the covariance into standard set of units → standardization.Standard deviation: a measure of the average deviation from the mean.If we divide any distance from the mean by the standard deviation, it gives us that distance in standard deviation units.We can express the covariance in a standard units of measurement if we divide it by...

StatisticsChapter 10Comparing two meansCategorical predictors in the linear modelIf we want to compare differences between the means of two groups, all we are doing is predicting an outcome based on membership of two groups.This is a linear model with one dichotomous predictor. Independent t-test: used when you want to compare two means that come from conditions consisting of different entities (this is sometimes called the independent-measures or independent-means t-test)Paired-samples t-test: also known as the dependent t-test. Is used when you want to compare two means that come from conditions consisting of the same or related entities.Rationale for the t-test Both t-tests have a similar rationale:two samples of data are collected and the sample means calculated. These might differ by either a little or a lotIf the samples come from the same population, then we expect their means to be roughly equal. Although it is possible for the means to differ because of sample variation, we would expect large differences between sample means to occur very infrequently. Under the null hypothesis we assume that the experimental manipulation has no effect on the participant’s behaviour: therefore, we expect means from two random samples to be very similar.We compare the difference between the sample means that we collected to the difference between the sample means that we would expect to obtain (in the long run) if there were no effect. We use the standard error as a gauge of the variability between sample means. If the standard error is small, then we expect most samples to have very similar means. When the standard error is large, large differences in sample means are more likely. If the...

StatisticsChapter 12Comparing several independent means ANOVA: analysis of variancethe same thing as the linear model or regression.In designs in which the group sizes are unequal, it is important that the baseline category contains a large number of cases to ensure that the estimates of the b-values are reliable.When we are predicting an outcome from group membership, predicted values from the model are the group means.If the group means are meaningfully different, then using the group means should be an effective way to predict scores.Predictioni = b0 + b1X + b2Y + ƐiControl = b0Using dummy coding ins only one of many ways to code dummy variables.an alternative is contrast coding: in which you code the dummy variables in such a way that the b-values represent differences between groups that you specifically hypothesized before collecting data.The F-test is an overall test that doesn’t identify differences between specific means. But, the model parameters do.Logic of the F-statisticThe F-statistic tests the overall fit of a linear model to a set of observed data.F is the ratio of how good the model is compared to how bad it is.When the model is based on group means, our predictions from the model are those means.if the group means are the same then our ability to predict the observed data will be poor (F will be small)if the means differ we will be able to better discriminate between cases from different groups (F will be large...

StatisticsChapter 14Factorial designs Factorial design: when an experiment has two or more independent variables.There are several types of factorial designs:Independent factorial design: there are several independent variables or predictors and each has been measured using different entities (between groups).Repeated-measures (related) factorial design: several independent variables or predictors have been measured, but the same entities have been used in all conditions.Mixed design: several independent variables or predictors have been measured: some have been measured with different entities, whereas others used the same entities.We can still fit a linear model to the design.Factorial ANOVA: the linear model with two or more categorical predictors that represent experimental independent variables. The general linear model takes the following general form:Yi =b0 + b1X1i+b2X2i+... +bnXni+ƐiWe can code participant’s category membership on variables with zeros and ones.For example:Attractivenessi = b0+b1Ai+b2Bi+b3ABi+Ɛib3AB is the interaction variable. It is A dummy multiplied by B dummy variable.Behind the scenes of factorial designs Calculating the F-statistic with two categorical predictors is very similar to when we had only one.We still find the total sum of squared errors (SST) and break this variance down into variance that can be...

StatisticsChapter 16Mixed designs Situations where we combine repeated-measures and independent designs.Mixed designs: when a design includes some independent variables that were measured using different entities and others that used repeated measures.A mixed design requires at least two independent variables.Because by adding independent variables we’re simply adding predictors to the linear model, you can have virtually any number of independent variables if your sample size is gin enough.We’re still essentially using the linear model.Because there are repeated measures involved, people typically use an ANOVA-style model. Mixed ANOVA All the sources of potential bias in chapter 6 apply.homogeneity of variancesphericityYou can apply the Greenhouse-Geisser correction and forget about sphericity. Mixed designs compare several means when there are two or more independent variables, and at least one of them has been measured using the same entities and at least one other has been measured using different entiteis.Correct for deviations from sphericity for the repeated-measures variable(s) by routinely interpreting the Greenhouse-Geisser corrected effects.The table labelled Tests of Within-Subject Effects shows the F-statistic(s) for any repeated-measures variables and all of the interaction effects. For each effect, read the row labelled Greenhouse-Geisser or Huynh-Feldt...

StatisticsChapter 18Exploratory factor analysisIn factor analysis, we take a lot of information (variables) and a computer effortlessly reduces this into a simple message (fewer variables). Latent variable: something that cannot be accessed directly.Measuring what the observable measures driven by the same underlying variable are.Factor analysis and principal component analysis (PCA) are techniques for identifying clusters of variables.Three main uses:To understand the structure of a set of variablesto construct a questionnaire to measure an underlying variableto reduce a data set to a more manageable size while retaining as much of the original information as possible. If we measure several variables, or ask someone several questions about themselves, the correlation between each pair of variables can be arranged in a table.this table is sometimes called the R-matrix.Factor analysis attempts to achieve parsimony by explaining the maximum amount of common variance in a correlation matrix using the smallest number of explanatory constructs.Explanatory constructs are known as latent variables (or factors) and they represent clusters of variables that correlate highly with each other.PCA differs in that it tries to explain the maximum amount of total variance in a correlation matrix by transforming the original variables into linear components.Factor analysis and PCA both aim...

Here is a short explanation how to do tests in SPSS. These are the tests needed for the third block of WSRt and psychology at the second year of the uva. Open the dataGo to analyse, correlate, bivariatePlace the variables of which you want to know the correlation under ‘variables’Click on ‘paste’ and run the syntax Open the dataGo to analyse, correlate, partialPlace the variable of which you want to know the correlation under ‘variables’Place the variable for which you want to control under ‘controlling for’Click on ‘options’Select ‘zero-order correlations’ (this is the correlation without controlling for one variable)Click on ‘continue’Click on ‘paste’ and run the syntax Open the dataGo to analyse, regression, linearPlace the dependent variable under ‘dependent’Place the independent variables under ‘independent’If you want to run more models, you can put the first variable under ‘independent’, click on ‘next’ and put the next variable under ‘independent’ (this...

Discovering statistics using IBM SPSS statisticsChapter 20Categorical outcomes: logistic regressionThis summary contains the information from chapter 20.8 and forward, the rest of the chapter is not necessary for the course. Logistic regression is a model for predicting categorical outcomes from categorical and continuous predictors.A binary logistic regression is when we’re trying to predict membership of only two categories.Multinominal is when we want to predict membership of more than two categories. The linear model can be expressed as: Yi = b0 + b1Xi + errorib0 is the value of the outcome when the predictors are zero (the intercept).The bs quantify the relationship between each predictor and outcome.X is the value of each predictor variable.One of the assumptions of the linear model is that the relationship between the predictors and outcome is linear.When the outcome variable is categorical, this assumption is violated.One way to solve this problem is to transform the data using the logarithmic transformation, where you can express a non-linear relationship in a linear way.In logistic regression, we predict the probability of Y occurring, P(Y) from known (logtransformed) values of X1 (or Xs).The logistic regression model with one predictor is:P(Y) = 1/(1+e –(b0 +b1X1i))The value of the model will lie between 1 and 0. ...