HPCS4031 Research Methods in Psychology Lecture 4: Experimental Design (2) PDF

HPCS4031 Research Methods in Psychology Lecture 4: Experimental Design (2) Between-Subjects (Independent Design) vs. Within-Subjects (Repeated-Measures Design) Between-subject design Participants in each experimental condition are independent of each other One-way design (only 1 IV)...

HPCS4031 Research Methods in Psychology Lecture 4: Experimental Design (2) Between-Subjects (Independent Design) vs. Within-Subjects (Repeated-Measures Design) Between-subject design Participants in each experimental condition are independent of each other One-way design (only 1 IV) Level 1 Level 2 Level 3 John Tim Peter Factorial design (more than 1 IVs) Within-subject design Participants will undergo all experimental conditions in a study One-way design (only 1 IV) Level 1 Level 2 Level 3 Factorial design (more than 1 IVs) John John John 2 Between-Subjects Design 3 Between-Subjects Design IV? DV? Number of levels? 4 Between-Subjects Design IV: Note taking method DV: Comprehension Number of levels: 2 5 Between-Subjects Design How to ensure the results are not confounded by participant variables? 6 Between-Subjects Design Random assignment: every person volunteering for the study has an equal chance of being placed in any of the conditions being formed Random assignment ≠ Random sampling Random sampling: a process designed to produce a sample of individuals that reflects the broader population Random assignment: a method for placing participants 7 Between-Subjects Design Pre-test: to show that both groups were similar in relevant performance before the experimental conditions were applied E.g., check that both groups were equivalent on a similar memory task before starting the experiment 8 Between-Subjects Design One-way design 1 IV with 2 or more levels 9 Between-Subjects Design E.g., Use of highlight pens → memory 2 levels of IV: highlight pen vs. no highlight pen. Highlight pen Memory test Random Assignment No highlight Memory Test pen 10 Between-Subjects Design E.g., Money → Happiness 3 levels of IV: No cash vs some cash vs A lot of cash Mood No cash assessment Random Mood Some cash Assignment assessment Mood A lot of cash assessment 11 Between-Subjects Design One-way design Examines the effect of different IV levels on the DV Allows us to explore the effect from a single factor; we call this a between-subject factor A between-subject factor does not necessarily involve random assignment. A between-subject factor can exist in a quasi-experiment too. E.g., effect of academic major on GPA. 12 Between-Subjects Design One-way design 13 Between-Subjects Design Strengths Weaknesses 14 Between-Subjects Design Strengths Each subject enters the study fresh, and naive with respect to the hypotheses to be tested → Prevent order effect Learning/practice effects Fatigue effects Shorter duration of experiment Weaknesses Large numbers of participants required The Differences between the conditions might be due to differences between the individuals in the different groups → threaten internal/external validity 15 Between-Subjects Factorial Design Factorial design: Any study with more than 1 Independent Variable, each with 2 or more levels 16 Between-Subjects Factorial Design (Case 1) Factorial design: E.g., suppose you wish to find out: (1) if recall can be improved by training people to use visual imagery while memorizing a list of words IV? DV? 17 Between-Subjects Factorial Design (Case 1) Factorial design: E.g., suppose you wish to find out: (1) if recall can be improved by training people to use visual imagery while memorizing a list of words IV: Visual imagery technique DV: Memory 18 Between-Subjects Factorial Design (Case 1) Factorial design: E.g., suppose you wish to find out: (1) if recall can be improved by training people to use visual imagery while memorizing a list of words Operational definition: IV: Visual imagery technique (experimental group) vs Rote repetition (control group) DV: No. of words memorised 19 Between-Subjects Factorial Design (Case 1) Factorial design: E.g., suppose you wish to find out: (2) how memory is affected by how quickly words are presented, that is, a word list’s presentation rate IV? DV? 20 Between-Subjects Factorial Design (Case 1) Factorial design: E.g., suppose you wish to find out: (2) how memory is affected by how quickly words are presented (i.e., a word list’s presentation rate) IV: Word list’s presentation rate DV: Memory 21 Between-Subjects Factorial Design (Case 1) Factorial design: E.g., suppose you wish to find out: (2) how memory is affected by how quickly words are presented (i.e., a word list’s presentation rate) Operational definition: IV: Presentation rate 2-sec/word (experimental group) vs 4-sec/word (control group) DV: No. of words memorised 22 Between-Subjects Factorial Design (Case 1) Factorial design: E.g., suppose you wish to find out: (1) if recall can be improved by training people to use visual imagery while memorizing a list of words (2) how memory is affected by how quickly words are presented, that is, a word list’s presentation rate → factorial design to find the answer for both questions 23 Between-Subjects Factorial Design (Case 1) Factorial design: Memory study: 2 X 2 design factorial matrix 24 Between-Subjects Factorial Design (Case 1) Factorial design: Memory study: 2 (imagery vs rote) X 2 (2-sec vs 4-sec) design In factorial design, levels ≠ conditions 4 conditions but not 2! 25 Between-Subjects Factorial Design (Case 1) Factorial design: If you want to study the effect of gender as well… 2 (Imagery vs Rote) X 2 (2-sec vs 4-sec) X 2 (Men vs Women) 26 Between-Subjects Factorial Design (Case 2) Factorial design E.g., 2x2 factorial design: 2 IVs, each IV has 2 levels 4 experimental conditions Be careful: Don’t read 2X2 as “there are 2 variables with 2 levels” 2X2 means “variable A has 2 levels and variable B has 2 level”. Course Mode (IV #2) Online Face to Face Major Psychology 2.4 3.6 (IV #1) Science 3.5 3.5 27 Between-Subjects Factorial Design (Case 2) Course Mode (IV #2) Online Face to Face Hybrid Psychology 2.4 3.6 3.0 Major (IV #1) Science 3.5 3.5 3.5 28 Between-Subjects Factorial Design (Case 2) Campus (IV #1) A B Course Mode (IV #3) Course Mode (IV #3) Online Face Hybrid Online Face Hybrid to to Face Face Major Psych 2.4 3.6 3.0 Major Psych 3.4 2.6 3.0 (IV (IV #2) Science 3.5 3.5 3.5 #2) Science 3.5 3.5 3.5 2 (A vs B) X 2 (psych. vs science) X 3 (online vs face to face vs hybrid) 29 How do we interpret the numbers in a Between-subjects factorial design? There are two kinds of effects: Main effects and interaction effects Main effects: the overall influence of each of the independent variables Interaction effects: whether the variables combine to form a more complex result 30 Between-Subjects Factorial Design – Interpretation (Case 1) First main effect: Main effect of training (Imagery vs Rote) The main effect of type of training is determined by combining the data for participants trained to use imagery (for both presentation rates combined) Comparing it to all of the data for participants using rote repetition 31 Between-Subjects Factorial Design – Interpretation (Case 1) Second main effect: Main effect of presentation rate (2-sec/item vs 4- sec/item) Combining the data for everyone presented the words at a 2‐second rate and comparing that with the data from those presented the words at a 4‐second rate 32 Between-Subjects Factorial Design – Interpretation (Case 1) Let’s consider hypothetical data for a memory experiment like the example we’ve been using. Assume 25 subjects in each condition (i.e., each cell of the matrix). Their task is to memorize a list of 30 words. The average number of words recalled for each of the four conditions might look like this: 33 Between-Subjects Factorial Design – Interpretation (Case 1) Does imagery training produce better recall than rote repetition? That is, is there a main effect of type of training? Mean of “imagery” group = ? Mean of “Rote” group = ? 34 Between-Subjects Factorial Design – Interpretation (Case 1) Does imagery training produce better recall than rote repetition? That is, is there a main effect of type of training? Mean of “imagery” group = (17 + 23)/2 = 20 Mean of “Rote” group = (12+18)/2 = 15 35 Between-Subjects Factorial Design – Interpretation (Case 1) Main effect of presentation rate? Mean of “2-sec” group = ? Mean of “4-sec” group = ? 36 Between-Subjects Factorial Design – Interpretation (Case 1) Main effect of presentation rate? Mean of “2-sec” group = (17+12)/2 = 14.5 Mean of “4-sec” group = (23+18)/2 = 20.5 37 Between-Subjects Factorial Design – Interpretation (Case 1) For these data, it appears that imagery improves memory (20 > 15) and that recall is higher if the words are presented at a slower rate (20.5 > 14.5). That is, there seem to be two main effects here Nevertheless, it takes an analysis of variance (ANOVA) to make a judgment about whether the differences are significant statistically or due to chance 38 Between-Subjects Factorial Design – Interpretation (Case 2) Let’s go back to this easier 2X2 design: Course Mode (IV #2) Online Face to Face Major Psychology 2.4 3.6 (IV #1) Science 3.5 3.5 In this 2X2 design, there are 2 main effects and 1 interaction effect: Main effect of Major; Main effect of Course Mode Major X Course Mode interaction effect 39 What are Main Effects? You look at the effect of one factor, while ignoring the effect of other factors. E.g., Main effect of Major Course Mode (IV #2) Online Face to Face Major Psychology 2.4 3.6 Mean =3.0 (IV #1) Science 3.5 3.5 Mean =3.5 Is there a significant difference between 3.0 vs. 3.5? If there is, then we say “there is a main effect of Major”. If there is not, then we say “the main effect of Major is insignificant” Later you will learn about the statistical test necessary to determine if the difference is statistically significant. 40 What are Main Effects? E.g., Main effect of Course Mode Course Mode (IV #2) Online Face to Face Major Psychology 2.4 3.6 (IV #1) Science 3.5 3.5 Mean = 2.95 Mean = 3.55 Is there a significant difference between 2.95 vs. 3.55? If there is, then we say “there is a main effect of Major”. If there is not, then we say “the main effect of Major is not significant” 41 Interaction Effect The distinct advantage of factorials over single‐factor designs lies in their potential to show interactive effects In a factorial design, an interaction is said to occur when the effect of one independent variable depends on the level of another independent variable. 42 Interaction Effect Suppose we hypothesize that an introductory psychology course is best taught as a laboratory self‐discovery course rather than as a straight lecture course, but we also wonder if this is generally true or true only for certain kinds of students. Perhaps science majors would especially benefit from the laboratory approach. 43 Interaction Effect To test the idea, we need to compare a lab with a lecture version of introductory psychology, but we also need to compare types of students, perhaps science majors and humanities majors. This calls for a 2 X 2 design that looks like this: 44 Interaction Effect Dependent variable: measure of learning (e.g., exam score) 45 Interaction Effect Any main effects? No—all of the row and column means are the same: 75 So did anything at all happen in this study? Yes—something clearly happened. Specifically, the science students did better in the lab course than in the lecture course, but the humanities students did better in the lecture course than in the lab course. 46 Interaction Effect The effect of one variable (course type) depended on the level of the other variable (major) Hence, even if no main effects occur, an interaction can occur and produce an interesting outcome 47 Interaction Effect This teaching example also highlights the distinct advantage of factorial designs over single factor designs. Suppose you completed the study as a single‐factor, two‐level design, comparing lab with lecture versions of introductory psychology. You would probably use a matched group design, with student GPA and perhaps major as matching variables. In effect, you might end up with the same people who were in the factorial example. 48 Interaction Effect However, by running it as a single‐factor design, your results would be: Lab course : 75 Lecture course : 75 You might conclude it doesn’t matter whether introductory psychology includes a lab or not. 49 Interaction Effect With the factorial design, however, you know the lab indeed matters, but only for certain types of students. In short, factorial designs can be more informative than single‐factor designs 50 What is an Interaction Effect? The effect of one IV on the DV depends on the level of another IV How do the factors combine to influence the DV? Sometimes, 2 factors individually may have no effect on the DV (i.e., no significant main effect). But when you combine them together, they interact to affect the DV E.g., You drink coke, nothing bad happens (no main effect of coke). You take a sleeping pill, nothing bad happens (no main effect of pill). But if you take them together, coke and sleeping pill can interact and make you feel sick. This is an interaction effect. 51 What is an Interaction Effect? Other times, there could be both main effects and interaction effects. In this case, the main effects become meaningless, and the interaction effect is what you should focus on Let’s go back to our GPA example to show you why 52 What is an Interaction Effect? Course Mode (IV #2) Online Face to Face Major Psychology 2.4 3.6 Mean = 3.0 (IV #1) Science 3.5 3.5 Mean = 3.5 Mean = 2.95 Mean = 3.55 Let’s suppose the main effects are significant If you just look at the main effect of Major, you might conclude that Science students are better than Psychology students. But is that really true? Is the story that simple? If you just look at the main effect of Course Mode, you might conclude that Online teaching is garbage. But is that the best interpretation? Is the story that simple? 53 What is an Interaction Effect? Course Mode (IV #2) Online Face to Face Major Psychology 2.4 3.6 Mean = 3.0 (IV #1) Science 3.5 3.5 Mean = 3.5 Mean = 2.95 Mean = 3.55 If there is a significant interaction effect, the aforementioned main effects become “qualified”. The correct interpretation should be: Science students are better than Psychology students, but only when the teaching mode is online (3.5 vs. 2.4). During normal f2f teaching, the two groups are almost the same. (3.5 vs. 3.6) F2f teaching is more effective than online teaching but only among Psychology students (3.6 vs. 2.4). Among Science students, there is no difference (3.5 vs. 3.5). The comparisons in parentheses are called “simple effects”. 54 Interaction Effect “Science students are better than Psychology students, but only when the teaching mode is online. During normal f2f teaching, Science and Psychology student are almost the same. “ “F2f teaching is more effective than online teaching but only among Psychology students. Among Science students, there is no difference.” Notice how these statements always consider both Major and Course Mode at the same time. If you have a significant interaction effect, you cannot just look at one factor (i.e., main effect) and pretend the other factor does not exist. Otherwise you’ll end up with overly simplistic/ incorrect conclusions. 55 Interaction Effect (one more example) Research question: Which drug is more effective? 2 (Long vs short) X 2 (Drug A vs Drug B) 56 Interaction Effect (one more example) 1st Main effect: Length of depression Long-term depression score = 8.22; Short-term score = 6.56 Those with short-term depression were less depressed at the end of the experiment than those with long-term depression 57 Interaction Effect (one more example) 2nd Main effect: Drug type Drug A = 6.39; Drug B = 8.38 Drug A is more effective than Drug B in general without regard to how long participants have been depressed 58 Interaction Effect (one more example) Is the effectiveness of the drugs dependent, in part, on the length of depression? By examining the individual cell means, it becomes clear that the answer is “yes.” Drug A is more effective for short-term than for long-term depression (4.67 vs. 8.11), while Drug B is about equally effective for both types of depression (8.32 vs. 8.45) 59 Interaction Effect (one more example) Implication of the interaction: If an individual has short-term depression, Drug A is indicated But if an individual has long-term depression, either drug is likely to be about equally effective. Thus, the two classification variables interact: The best drug to take is dependent on the length of the depression. 60 More Factors = Better study? In a 2X2X2 design, there are 3 main effects (because there are 3 factors) and 4 interaction effects (3 two-way interactions + 1 three-way interaction). Main effect of Major; Main effect of Course Mode; Main effect of Campus Major X Course Mode; Major X Campus; Course Mode X Campus; Major X Course Mode X Campus As you can see, things can get really messy quickly. 2 and 3-factor models are common but 4+ factors are rare. 61 Recognizing INTERACTION EFFECTs There is a quick and easy way to GPA by Major X Course Mode 4 recognize a potential interaction 3.5 effect 3 Plot the all the group means on a GPA 2.5 2 line graph and look for non- 1.5 parallel lines: 1 Science Non-parallel lines likely indicate an 0.5 Psychology interaction effect (pending statistical 0 tests, of course) Online Face to Face 62 Recognizing INTERACTION EFFECTs The effect of teaching mode on GPA by Major X Course Mode 4 GPA depends on the major 3.5 For science students, both modes 3 are the same 2.5 For psychology students, face-to- GPA 2 face teaching is preferred Science 1.5 Psychology 1 0.5 0 Online Face to Face 63 Recognizing INTERACTION EFFECTs 64 What about when there is no interaction effect? Then you will see lines that are relatively parallel to each other. Main effect of Course Mode only Main effect of Major only Main effects of both Major & Coruse Mode, but 4 4 no interaction 3.5 3.5 4.5 3 3 4 3.5 2.5 2.5 3 2 2 2.5 1.5 1.5 2 1.5 1 1 1 0.5 0.5 0.5 0 0 0 Online Face to face Online Face to face Online Face to face Psychology Science Psychology Science Psychology Science 65 Test your understanding Main effect for the cell phone? Main effect for age? Age x Cell phone interaction? 66 Test your understanding Main effect for the cell phone: No Main effect for age: No Age x Cell phone interaction: Yes, younger drivers on a cell phone cause more accidents; older drivers on a cell phone cause fewer accidents 67 Test your understanding Main effect for the cell phone? Main effect for age? Age x Cell phone interaction? 68 Test your understanding Main effect for the cell phone: No Main effect for age: Yes—younger drivers have more accidents Age x Cell phone interaction: No 69 Test your understanding Main effect for the cell phone? Main effect for age? Age x Cell phone interaction? 70 Test your understanding Main effect for the cell phone: Yes – cell phones cause more accidents Main effect for age: No Age x Cell phone interaction: No 71 Test your understanding Main effect for the cell phone? Main effect for age? Age x Cell phone interaction? 72 Test your understanding Main effect for the cell phone: Yes – cell phones cause more accidents Main effect for age: Yes—younger drivers have more accidents Age x Cell phone interaction: for younger drivers, cell phones do not make a difference, but for older drivers, cell phones cause more accidents 73 Within-Subjects Design 74 Within-Subjects Design (or Repeated – Measures) Each participant undergoes all experimental conditions E.g., The effect of Font Size (IV) on reading speed (DV) IV: Font Size, 3 levels (small, medium, large) DV: Reading speed (number of words they can read in 5 minutes) Participants are required to read a document with 3 different font sizes. 75 Within-Subjects Design (or Repeated – Measures) Research by Boothby, Clark, and Bargh (2014) Research question: Whether a shared experience would be intensified even when people do not interact with the other person H1: Shared experiences are amplified compared with unshared experiences H0: There is no difference between shared and unshared experiences 76 Within-Subjects Design (or Repeated – Measures) Research by Boothby, Clark, and Bargh (2014) 23 college women were recruited to a laboratory. Each participant was joined by a female confederate. The two sat side-by-side, facing forward, and never spoke to each other. Task: tasting some dark chocolates or viewing some paintings Participant: Chocolate → Chocolate Confederate: Chocolate → Paintings 77 Within-Subjects Design (or Repeated – Measures) Research by Boothby, Clark, and Bargh (2014) The participant was told that the two chocolates were different, but in fact they were exactly the same. After tasting each chocolate, participants rated how much they liked it 78 Within-Subjects Design (or Repeated – Measures) Within-subject experiments can be also manipulated in one-way and factorial design. Within-subject (one-way) IV Lv1 IV Lv2 IV Lv3 E.g., IV: Character size (3 levels) John John John Within-subject (factorial) E.g., IV1 Level 1 IV1 Level 2 IV1: Character size (2 levels) John John IV2: Types of background noise IV2 Level 1 IV2 Level 2 (2 levels; music vs. conversation) John John 79 Within-Subjects Design (or Repeated – Measures) Advantages 80 Within-Subjects Design (or Repeated – Measures) Advantages Subject variables become controlled variables Participants in each group become more “similar” (actually they are exactly the same) → ensures the participants in the two groups will be equivalent Minimize the selection bias 81 Within-Subjects Design (or Repeated – Measures) Advantages Greater statistical power than between-subject design Power: the probability to detect a statistically significant difference among experimental conditions Statistically speaking, when extraneous differences (unsystematic variability) in personality, food preferences, gender, ability, and so on are held constant across all conditions, researchers will be more likely to detect an effect of the independent variable manipulation if there is one E.g., mindfulness training on GRE scores → Hard to find the difference if extraneous differences exist between two groups, too much unsystematic variability may be obscuring a true difference 82 Within-Subjects Design (or Repeated – Measures) Advantages Fewer participants are needed (e.g., John, John, John vs. John, Tim, Peter from Slide 2) 83 Within-Subjects Design (or Repeated – Measures) Advantages High internal validity No selection effects: Because participants are exactly the same in the two conditions Recap: Selection effect – participants in one level of the independent variable are systematically different from those in the other Design confound is well controlled by ensuring all participants receive the same treatment 84 Within-Subjects Design (or Repeated – Measures) Disadvantages 85 Within-Subjects Design (or Repeated – Measures) Disadvantages Order effects: Being exposed to one condition changes how participants react to the other condition Practice/fatigue/boredom effects: a long sequence might lead participants to get better at the task, or to get tired or bored toward the end Carryover effects: some form of contamination carries over from one condition to the next For example, in experiments in which participants have to estimate the weight of objects they are likely to be influenced by how heavy the previous object was that they estimated 86 Within-Subjects Design (or Repeated – Measures) Disadvantages Order effects E.g., Background effect on reading performance I.V. Background noise: music (condition 1) VS conversation (condition 2) D.V. Reading speed Solution: Counterbalancing 87 Within-Subjects Design (or Repeated – Measures) Disadvantages Greater demand characteristics Demand characteristics: A cue that can lead participants to guess an experiment’s hypothesis Higher chance for participants to figure out the experiment objectives Participants may change behaviors to be more favorable/unfavorable to the experimenters (observer bias) Observer bias: occurs when observers’ expectations influence their interpretation of the participants’ behaviours or the outcome of the study Solution: Counterbalancing/Mixed trials presentation 88 Counterbalancing Counterbalancing Presenting the levels of the independent variable to participants in different sequences, i.e., each subject is exposed to every possible combination of experimental conditions With counterbalancing, any order effects should cancel each other out when all the data are collected 89 Complete/FULL Counterbalancing Participants are randomly divided into subgroups of equal size. Participants of each subgroups get the treatments in one of the possible orders (i.e., order is actually a between-subject factor, if you think about it) 2 conditions—“ABBA” counterbalance Example: A (music), B(conversation) Subgroup 1: AB Subgroup 2: BA The order itself becomes a factor. Usually, you are hoping that order has no effect. 90 Complete/FULL Counterbalancing 2 conditions—“ABBA” counterbalance Example: A (shared), B(unshared) Subgroup 1: AB Subgroup 2: BA A B B A 91 Example 350 Hypothesis: People read faster when the background noise is 300 music rather than conversation. Ideally, you would want to see 250 that this hypothesis is true 200 regardless of order. In this case, the hypothesis is 150 Music - Conversation - supported. Regardless of Conversation Music whether it is first or second, 100 Music 300 310 mean = 305 music causes people to read 50 Conversation 200 210 mean = 205 faster than when the mean = 250 mean = 260 background noise is 0 conversation Music - Conversation Conversation - Music Music Conversation 92 Example 600 However, when there is an 500 Order (main or interaction) effect … 400 The internal validity of our finding is now threatened by 300 an order effect. convo 200 music Music - Conversation - Music Conversation 100 Music 500 200 mean = 350 Conversation 150 250 mean = 200 mean = 325 mean = 225 0 Music - Conversation Conversation - Music Music Conversation 93 Complete/FULL Counterbalancing Suppose you add a third level to the background noise factor: Street noise A = Music, B= Conversation, C = Street noise 3 treatments — number of orders = 3! = 6 different groups are needed for complete counterbalancing ABC, ACB, BAC, BCA, CAB, CBA ! means (3X2X1); it is read as ”factorial”, as in ”3-factorial” Problem: 4! = (4X3X2X1) = 24 groups; 5! = (5X4X3X2X1) = 120 groups…complete counterbalancing becomes impossible. 94 Incomplete Counterbalancing Incomplete Counterbalancing / Regular Latin Square: each treatment occurs equally often in each portion of the experiment. (i.e., Condition A occurs first, second, and third equally often, so do conditions B and C) A Latin Square can help us do that: A 4! example Treatment Order 1st 2nd 3rd 4th Participant 1 A B C D Participant 2 B C D A Participant 3 C D A B Participant 4 D A B C Note: If you have 4 treatments, then you at least need 4 participants. 95 Incomplete Counterbalancing A Latin square for six conditions (conditions 1 through 6) looks like this: 1 2 6 3 5 4 2 3 1 4 6 5 3 4 2 5 1 6 4 5 3 6 2 1 96 Incomplete Counterbalancing A Latin square for six conditions (conditions 1 through 6) looks like this: 1 2 6 3 5 4 2 3 1 4 6 5 3 4 2 5 1 6 4 5 3 6 2 1 5 6 4 1 3 2 97 Incomplete Counterbalancing A Latin square for six conditions (conditions 1 through 6) looks like this: 1 2 6 3 5 4 2 3 1 4 6 5 3 4 2 5 1 6 4 5 3 6 2 1 5 6 4 1 3 2 6 1 5 2 4 1 The first row is set up according to a formula, and then the conditions simply go in numerical order down each column 98 Incomplete Counterbalancing Balanced Latin Square: each treatment comes before and follow very other treatment equally often. To achieve this, you need to fill in the squares in the first row following this formula = 1, 2, n, 3, n-1, 4, n-2,… A B D C n=? B C A D C D B A 1 = A, 2 = B, 3 = C, 4 = D D A C B In other words, A, B, D, C For each subsequent row, add 1 to the formula. Note: can only be done for even n. 99 Incomplete Counterbalancing Balanced Latin Square: each treatment comes before and follow very other treatment equally often. To achieve this, you need to fill in the squares in the first row following this formula = 1, 2, n, 3, n-1, 4, n-2,… A B D C n = no. of conditions B C A D 1 = A, 2 = B, 3 = C, 4 = D C D B A In other words, A, B, D, C D A C B For each subsequent row, add 1 to the formula. Note: can only be done for even n. 100 A Note on Counterbalancing Complete counterbalancing if the number of conditions is low (I.e.,