Statistics for the Behavioural Sciences

An Introduction to Frequentist and Bayesian Approaches

Riccardo Russo

• Publisher: Routledge
• Year: 2020
• ISBN:
  9781351778985
• Edition: 2
• Topic: Psychology
• Subtopic: History & Theory in Psychology
• Index: Psychology

This accessible textbook is for those without a mathematical background (just some notions of basic algebra are sufficient) and provides a comprehensive introduction to all topics covered in introductory behavioural science statistics courses. It includes plenty of real examples to demonstrate approaches in depth based on real psychology experiments utilizing the statistical techniques described.

New content in this thoroughly updated second edition includes an introduction to Bayesian statistics which complements the coverage of Classical/Frequentist statistics present in the first edition. It also offers practical details on how to perform analyses using JASP – a globally employed, freely downloadable statistical package. The updated eResources also feature a range of new material including additional exercises so readers can test themselves on what they have learned in the book.

This timely and highly readable text will be invaluable to undergraduate students of psychology and research methods courses in related disciplines, as well as anyone with an interest in understanding and applying the basic concepts and inferential techniques associated with statistics in the behavioural sciences.

1 Introduction and basic concepts

1.1 Why is statistics useful in the behavioural sciences?

To say the least, it is quite common that a large number of students starting an undergraduate degree in psychology are not enthusiastic about having to study statistics and research methodology. It is highly possible that this lack of enthusiasm originates, at least in part, from the unappreciation of the relevance that statistics plays in the scientific understanding of human behaviour. The aim of this introduction is to provide a convincing argument, using a simple example, as to why some statistical knowledge is useful to students in the behavioural sciences.

1.2 Simple example of statistical testing

While revising for the end of year exams a university student notices that the material is better remembered when some classical music is played in the background, while less information is retained when no background music is played. This observation is used by this student as a basis to support their theory that listening to classical music while studying facilitates the memorisation of the exam material. As a consequence, he/she suggests the classmates to adopt the same strategy when revising. If you were one of the classmates would you be so impressed by your friend's theory and follow the suggestion? Probably not, but why not?

One objection that could be raised is that classical music may work for your friend, but its effect may not generalise to other people. After all, not everybody likes classical music; so there is the possibility that some students may find that this music interferes with their studying. Therefore, in order to see if the effect of listening to classical music on memory for exam material is not specific to your friend, a relatively large number of students should be tested. A second objection refers to the way in which the assessment of the two methods of study was made. More precisely what kind of evidence was used to claim that the music method led to better retention than the no music method? It is likely that your classmate had a feeling that they remembered more information after listening to classical music, but no formal assessment was made. However, it would be appropriate to have a more accurate way of measuring the amount of information retained under the two study conditions. Basically, it is important to obtain some sort of measurement of the phenomenon under investigation.

Overall it appears the theory suggesting that listening to classical music while studying helps memorisation could potentially be correct, but the evidence used to support this theory is not compelling. Hence, more people should be tested, and a clear way to measure the amount of information retained is fundamental to assess if the effect of listening to classical music while studying leads to an increment in the amount of information memorised compared to a condition where no music is played.

Thus, let us imagine implementing a study to provide a better evaluation of what we could name the “classical music theory”. As remarked above, to have a clearer assessment of the effect of classical music on memory for exam material, we will need to test a sizeable number of people (how large this number should be is discussed in later chapters when statistical power will be introduced). At this point a further decision needs to be made. Should the same group of people be tested in both study conditions (i.e., with and without exposure to classical music)? Or should two different groups of people be used so that, within each group, people are tested in only one of the two conditions? These two approaches have both advantages and disadvantages that will be discussed later in the book. For the moment, and without questioning why, we use the second approach. Therefore, one group of students will be given some material to study for a given amount of time during which classical music will be played in the background. A second group will receive the same material to be studied in the same amount of time, but no music will be played in the background. If classical music is beneficial we should find that the group exposed to classical music should retain more information than the group not exposed to music while studying.

Before discussing how we can measure memory for the material studied, some issues about the selection of the people participating in the study need to be addressed. Participants should be allocated to each of the two groups in a way that no bias is introduced in the evaluation of the two study techniques. Assuming, for simplicity, that 30 university students participate in the study, it would be inappropriate, for example, for the best students to be allocated to the music condition while the poorest students to be allocated to the no music condition. Under these circumstances it is likely that the classical music group would remember more information than the no classical music group simply because the students in the music group are more capable, and not because of the music factor. A study like this would lack internal validity, i.e., its results do not represent what we think they should represent. A way to circumvent the above problem would be to randomly allocate the participating students to the two conditions of the study, and, possibly, to have the same number of students in each group. In doing this, roughly the same number of able and poor students should, in principle, be included in each group so that these individual differences should be comparable between the two groups, and, therefore, students' ability should not bias the result of the study. Random allocation is fundamental because it reduces the risk that there is a confound between students' ability and the conditions of the study (e.g., that the better students are selectively allocated to the classical music group).

Before discussing how to measure students' memory for the exam material, a further digression on the selection of the participants to the study is in order. We said earlier that 30 university students should be selected to take part in the study. Thus we study a sample, and not the entire population of university students, because it would be unrealistic, very costly, and time consuming to test the entire population. However, we would like the results obtained with the selected sample to generalise to the entire relevant population. In order to generalise the results of our study to the entire population of university students, our selected sample should accurately reflect the characteristics of the entire population. If this occurs, it is then said that the study conducted has external validity. However, if a sample is biased, then the results obtained are biased too, thus these cannot be generalised to the entire population being studied. For example, if all of our selected students are very able and all likely to obtain “A” marks in their exams, we will have a biased sample because only a small percentage of students perform so well. The optimal way to obtain an unbiased sample is to draw a random sample from the entire population (using procedures similar to those used in the National Lottery draw). In this way every member of the population has equal probability of being included in the sample, thus the sample obtained should accurately reflect the characteristics of the population of university students. However, random sampling from the entire population is not practically possible in most real research. Nevertheless, it is important that samples are reasonably representative of the populations we want to generalise to (for more details on sampling procedures see Upton & Cook, 1997). Obviously, the extent to which the sample obtained is not representative of the entire relevant population leads to a limitation in the generalisation of the results that can be obtained in a study. For example, if we study how families with a monthly income up to £1500 per capita spend their money, which we may have difficulty in generalising the results to the population of families with a monthly income up to £3000 per capita.

Summarising, to test any psychological theory or hypothesis we need to study and measure some behaviour in a sample of subjects, and in doing this we aim to draw conclusions on the entire population from which the sample is taken. Therefore, while studying the behaviour of a relatively small sample we want to be able to generalise the results obtained to the entire relevant population. As an addendum it should be said that populations are not necessarily finite. A population could be the collection of a potentially infinite number of items (e.g., the set of all possible e-mails that people could potentially write). In this case, it would be impossible to list all the elements in the population.

Let us now consider how we can measure the retention of the studied material. For example, each group of 15 students have to study a prose passage for 1 h, the content of which is comparable to exam material where 60 important pieces of information are highlighted. Remember that we sampled 30 students, of which 15 to be exposed to classical music while studying and 15 not exposed to any background music. A convenient index, but not necessarily the best one to measure the retention of the to-be-learned information, is the number of key pieces of information correctly recalled 24 h after the study phase. Hence, the relevant information we will obtain is going to be the “number of key pieces of information retrieved by each student”, and this measure will be used to decide if students retain more information when they listen to classical music while studying compared to a no background music condition.

A reasonable approach to decide on the above matter would be to suggest that classical music is beneficial if the students in the “classical music” group recalled more units of information than the students in the “no music” group. Unfortunately, at first, the obtained data will only be an unorganised set of 15 memory scores for each of the two groups of students who participated in the study, and no useful information is likely to be extracted from these numbers without further manipulations. Some order needs to be imposed on these data to be able to decide which of the two groups of students recalled more information. For example, as a first step, the performance of the students in each group could be ranked in order. This would permit the inspection of the memory scores for each group, ordered from the lowest to the highest. A useful piece of information is also the average number of items recalled in each group (see Table 1.1a). A quick glance at the rank ordered data seems to suggest that more items are recalled in the classical music group (i.e., for almost any rank more items are recalled in the classical music than in the no music group). Moreover, on average, more items are recalled in the classical music group. It seems, therefore, we could answer the original question and conclude that, when classical music is played while studying, more information about exam material is remembered compared to a condition when no background music is played.

Table 1.1a Fictitious data on the effect of listening to classical music while studying on the retention of the material to be learned. In Table 1.1a data are ordered within each studying condition (where I correspond to the rank of the lowest scoring student and XV to the highest). In Table 1.1b data are ordered irrespective of the studying condition

Table 1.1b

This conclusion may, however, be premature. In fact, various reasons suggest that the type of inference used above to decide on the memory effect of classical music is incorrect. First of all, taking a closer look at the rank ordered data might indicate that the performances in the two groups are comparable. Apart from a small number of relatively extreme scores, which appear to be larger in the classical music group, the majority of scores are comparable in the two groups (see Table 1.1b where all the scores are ordered from the smallest to the largest).

The second important point to make is that, despite the average memory scores being numerically different, this difference could have occurred simply by chance and not because the two groups in the study were treated differently. To illustrate this concept, imagine that 30 more students are randomly sampled from the same population of university students, and that these students are randomly assigned to two groups. These are conveniently labelled Group A and Group B. However, unlike the previous study, both groups are now asked to commit to memory the relevant material while there is no music in the background for either group. Table 1.2 displays the rank ordered performances of the two groups. There are some noticeable similarities between the data displayed in Table 1.1a and Table 1.2. The most striking features are that in both tables the average performance of Group A is numerically larger than the performance of Group B, and that the average performance of the A groups is almost identical in the two tables. Thus the results obtained in the second study are very similar to those previously found despite no classical music ever being played in the background. Therefore, these results cast doubts on the validity of the previous conclusion regarding the effectiveness of background classical music in improving students' memory for exam material.

Table 1.2 Hypothetical replication of the study of the effect of listening to classical music on learning, where no music is played at all

How then is it possible to obtain a set of results like those just described? As previously mentioned, we assumed that the students taking part in both studies were randomly sampled from the population of university students, and that they were further randomly allocated to the A and B music groups. This process is fair since, in principle, it avoids introducing biases in the allocation of students to different groups. It is, however, important to keep in mind that students differ along several dimensions, as for example in their abilities to memorise. It then follows that, when students are randomly assigned to different groups, random variations in the average characteristics of the students in these different groups are bound to happen. Thus these random differences can influence memory performance even if the two groups are treated in the same way (i.e., no music is played during learning). As a consequence, what may have happened is that the two groups obtained different average memory scores simply by chance, despite all selected students being randomly sampled from the population of university students, and despite the students being randomly allocated to different groups. Students are not identical, so despite being randomly allocated, small differences in the recall abilities of different groups may have occurred simply by chance. In the second study, where both groups of students were treated in the same way, the small difference in the average scores must have occurred by chance. Given that the results in the second study are very similar to those obtained in the first study, it could well be that the numerical advantage in the average memory score obtained in the first study by the “real” classical music group had nothing to do with the effect of music, but simply reflected the fact that the average performances of random samples drawn from a given population differ by chance.

The above reasoning points out the need for an appropriate methodology to decide if classical music played in the background does indeed improve memory for exam information, or if the average numerical advantage shown by students exposed to classical music in the first study simply reflected random variations which occur naturally among samples of students randomly allocated from the population of university students. Inferential statistics techniques aim to provide an answer to these types of question. A large part of this book is dedicated to describing some of these techniques. Furthermore, notice that the process of statistical inference is a probabilistic one. Therefore, a chapter of this book will be dedicated to the description of some basic concepts of probability theory.

This introduction has gone a long way towards showing how various approaches, which can be classified within the field of statistics, are essential in the behavioural sciences. We started with a fairly simple and admittedly quite naive psychological theory. We then showed that to test this theory some measurement of the phenomenon under investigation was required, and that a relatively large sample of people needed to be tested under different study conditions. We discussed some implications involved in assigning subjects to study conditions, and the need to provide some preliminary description of the collected data. We then demonstrated the need for inferential statistics techniques to evaluate the plausibility of a psychological theory using the data collected in an empirical study. Finally, noticed an example above indicated that the need for a quite sophisticated approach to empirically test a question that appeared potentially interesting (i.e., the effect of music on retention). Research questions often arise to shed light on an interesting phenomenon. It is important to keep in mind that in this process an attempt should be made to provide a theoretical understanding of the phenomenon. It then follows that research should be conducted to test theoretically relevant questions, so that the outcome of the research either support...