Variance of a single variable represents the average amount that the data vary from the mean. The cross-product deviation multiplies the deviation for one variable by the corresponding deviation for the second variable. The average value of the cross-product deviation is the covariance. This is an averaged sum of combined deviation. It uses the following formula:A positive covariance indicates that if one variable deviates from the mean, the other variable deviates in the same direction. A negative covariance indicates that if one variable deviates from the mean, the other variable deviates in the opposite direction. Covariance is not standardized and depends on the scale of measurement. The standardized covariance is the correlation coefficient and is calculated using the following formula:A correlation coefficient of values 0.1 represents a small effect. Values of 0.3 represent a medium effect and values of 0.5 represent a large effect. In order to test the null hypothesis of the correlation, namely that the correlation is zero, z-scores can be used. In order to use the z-scores, the distribution must be normal, but the r-sampling distribution is not normal. The following formula adjusts r in order to make the sampling distribution normal:The standard error uses the following formula:This leads to the following formula for z:The null hypothesis of correlations can also be tested using the t-score with degrees of freedom N-2:The confidence intervals for the correlation uses the same formula as all the other confidence intervals. These values have to be converted back to a correlation efficient using the following formula:CORRELATIONNormality in correlation is only important if the sample size is small (1), there is significance testing (2) or there is a confidence interval (3). The assumptions of correlation are normality (1) and linearity (2). The correlation coefficient squared (R2) is a measure of the amount of...

Moderation refers to the combined effect of two or more predictor variables on an outcome. This is also known as an interaction effect. A moderator variable is one that affects the relationship between two others. It affects the strength or direction of the relationship between the variables.The interaction effect indicates whether moderation has occurred. The predictor and the moderator must be included for the interaction term to be valid. If, in the linear model, the interaction effect is included, then the individual predictors represent the regression of the outcome on that predictor when the other predictor is zero. The predictors are often transformed using grand mean centring. Centring refers to transforming a variable into deviations around a fixed point. This fixed point is typically the grand mean. Centring is important when the model contains an interaction effect, as it makes the bs for lower-order effects interpretable. It makes interpreting the main effects easier (lower-order effects) if the interaction effect is not significant. The bs of individual predictors can be interpreted as the effect of that predictor at the mean value of the sample (1) and the average effect of the predictor across the range of scores for the other predictors (2) when the variables are centred. In order to interpret a (significant) moderation effect, a simple slopes analysis needs to be conducted. It is comparing the relationship between the predictor and outcome at low and high levels of the moderator. SPSS gives a zone of significance. Between two values of the moderator the predictor does not significantly predict the outcome and below and above the values it does.The steps for moderation are the following if there is a significant interaction effect: centre the predictor and moderator (1), create the interaction term (2), run a forced entry regression with the centred variables and the interaction of the two centred variables (3). The simple...

Confounding bias occurs when a variable influences both who is selected for the treatment and the outcome of the experiment. If a possible confounding variable is known, it is possible to control for the possible confounding variable. Researchers tend to control for all possible variables, which leaves the possibility of controlling for the thing you are trying to measure (e.g. controlling for mediators).Confounding needs a causal solution, not a statistical one and causal diagrams provide a complete and systematic way of finding that solution. If all the confounders are controlled for, a causal claim can be made. However, it is not always sure whether all confounders are controlled for.Randomization has two clear benefits. It eliminates confounder bias and it enables the researcher to quantify his uncertainty. Randomization eliminates confounders without introducing new confounders. In a non-randomized study, confounders must be eliminated by controlling for them, although it is not always possible to know all the possible confounders. It is not always possible to conduct a randomized controlled experiment because of ethical, practical or other constraints. Causal estimates of observational studies can provide with provisional causality. This is causality contingent upon the set of assumptions that the causal diagram advertises.Confounding stands for the discrepancy between what we want to assess (the causal effect) and what we actually do assess using statistical methods. A mediator is the variable that explains the causal effect of X on Y (X>Z>Y). If you control for a mediator, you will conclude that there is no causal link, when there is. There are several rules for controlling for possible confounders:In a chain junction (A -> B -> C), controlling for B prevents information from A getting to C and vice versa.In a fork or confounding junction (A C), controlling for B prevents information from A getting to C...

Simpson’s paradox states that the direction of an association at the population-level may be reversed within subgroups of that population. Inadequate attention to the Simpson’s paradox may lead to faulty inferences. The Simpson’s paradox can arise because of differences in proportions on subgroup levels compared to population levels. It also states that a pattern (association) does not need to hold within a subgroup.The paradox is related to a lot of things, including causal inference. A generalized conclusion (e.g. extraversion causes party-going) might hold for the general population, but does not mean that this inference can be drawn at the individual level. A correlation across the population does not need to hold in an individual over time. In order to deal with Simpson’s paradox, the situations in which the paradox occurs frequently have to be assessed. There are several steps in preventing Simpson’s paradox:Consider when it occurs.Explicitly propose a mechanism, determining at which level it is presumed to operate.Assess whether the explanatory level of data collection aligns with the explanatory level of the proposed mechanism. Conduct an experiment to assess the association between variables.In the absence of strong top-down knowledge, people are more likely to make false inferences based on Simpson’s paradox....

One way of avoiding the null hypothesis testing ritual in science is to increase the precision of theories by casting them as formal models. Rituals can be characterized by a repetition of the same action (1), fixations on special features (2), anxieties about punishment for rule violation (3) and wishful thinking (4). The null hypothesis testing ritual is mainly maintained because many psychological theories are too weak to make precise predictions besides the direction of the effect.A model is a simplified representation of the world that aims to explain observed data. It specifies a theory’s predictions. Modelling is especially suited for basic and applied research about the cognitive system. There are four advantages of formally specifying the theories as models:Designing strong tests of theoriesModelling theories leads to being able to make quantitative predictions about a theory, which then leads to comparable, competing predictions between theories which allows for comparison and testing of theories.Sharpening research questionsNull hypothesis testing allows for vague descriptions of theories and specifying the theories as models requires more precise research questions. These vague descriptions make theories difficult to test and sharpening the research questions makes it easier to test the theories.Going beyond linear theoriesNull hypothesis testing is especially applicable to simple hypotheses. The statistical tools available are used to create theories, mostly linear theories and by specifying the theory as a model, this is not necessary anymore.Using more externally valid designs to study real-world questionsModelling can lead to more externally valid designs, as confounds are not eliminated in the analysis, but built into the model. Goodness-of-fit measures cannot make the distinction between variation in the data as a result of noise or as a result of the psychological process of interest. A model can end up...

There are four procedures to present the implications of a correlation in terms of our ability to use the correlations to make successful predictions:Binomial effect size display (dichotomous)This illustrates the practical consequences of using correlations to make decisions. It can show how many successful and unsuccessful predictions can be made on the basis of a correlation. It uses the following formula:Binomial effect size display can be used to translate a validity correlation into an intuitive framework. However, it frames the situation in terms of an ‘equal proportions’ situation. Taylor-Russell tables (dichotomous)These tables inform selection decisions and provide a probability that a prediction will result in a successful performance on a criterion. The size of the validity coefficient (1), selection proportion (2) and the base rate (3) are required for the tables. Utility analysisThis frames validity in terms of a cost-benefit analysis of test use. Analysis of test sensitivity and test specificityA test is evaluated in terms of its ability to produce correct identifications of a categorical difference. This is useful for tests that are designed to detect a categorical difference.Validity correlations can be evaluated in the context of a particular area of research or application.A nomological network refers to the interconnections between a construct and other related construct. There are several methods to evaluate the degree to which measures show convergent and discriminate associations:Focusses associationsThis method focusses on a few highly relevant criterion variables. This can make use of validity generalization. Sets of correlationsThis method focusses on a broad range of criterion variables and computes the correlations between the test and many criterion variables. The degree to which the pattern of correlations ‘makes sense’ given the conceptual meaning of the construct is evaluated.Multitrait-...

Scaling refers to assigning numerical values to psychological attributes. Individuals in a group should be similar to each other in the regard of sharing a psychological feature. There are rules to follow in order to put people in categories:People in a category must be identical with respect to the feature that categorizes the group (e.g. hair colour).The groups must be mutually exclusiveThe groups must be exhaustive (e.g. everyone in the population can fall into a category).Each person should fall into one category and not more than one. If numerals are used to indicate order, then the numerals serve as labels indicating rank. If numerals have the property of quantity, then they convey information about the exact amounts of an attribute. Units of measurement are standardized quantities. The three levels of groups are identity (1), order (2) and quantity (3).There are two possible meanings of the number zero. It can be the absolute zero (1) (e.g. a reaction time of 0ms) or it can be an arbitrary quantity of an attribute (2). This is called the arbitrary zero. The arbitrary zero does not represent the absence of anything, rather, it is a point on a scale to measure that feature. A lot of psychological attributes use the arbitrary zero (e.g. social skill, self-esteem, intelligence).An unit of measurement might be arbitrary because unit size may be arbitrary (1), some units of measurement are not tied to any one type of object (2) (e.g. centimetres can measure anything with a spatial property) and some units of measurement can be used to measure different features of the same object (3) (e.g. weight and length).One assumption of counting is additivity. This requires that unit size does not change. This would mean that an increase of one point is equal at every point. This is not always the case, as an IQ test asks increasingly difficult questions to increase one point of IQ. Therefore, the unit size changes.Counting only qualifies as...

Psi refers to the anomalous retroactive influence of future events on an individual’s current behaviour. There are three important deficiencies in modal research practice: an overemphasis on conceptual replication (1), insufficient attention to verifying the integrity of measurement instruments and experimental procedures (2) problems with the implementation of null hypothesis testing (3). The interpretation bias refers to a bias towards interpretations of data that favour a researcher’s theory. A potential consequence of this is an increased risk of reported false positives and a disregard of true negatives. The knowledge system of psychology consists of theory relevant beliefs (1), this is about the mechanisms that produce behaviour and method-relevant beliefs (2), this is about the procedures through which data is obtained. Deficiencies in modal research practice bias systematically bias the interpretation of confirmatory data as theory relevant (1) and the interpretation of disconfirmatory data as method relevant (2).Central beliefs are beliefs on which many other beliefs depend. Conservatism refers to choosing the theoretical explanation consistent with the data that requires the least amount of restructuring of the existing knowledge system. If method-relevant beliefs are central in a knowledge system, it becomes more difficult to blame methodology related errors for disconfirmatory results. If theory-relevant beliefs become central, it poses the threat of becoming a logical assumption. A hypothesis under test should be described in a way that is falsifiable and not logically necessary. An overemphasis on conceptual replication at the expense of direct replication weakens method-relevant beliefs in the knowledge system. A statistical significant result is often followed by a conceptual replication. A failure of the conceptual replication leads to the question whether the negative result was due to the falsity of the underlying theory...