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PSYM021 Introduction to Methods & Statistics Week Three: Statistical techniques I Cris Burgess Reporting statistics It isn’t enough to insert a table and leave it at that You must follow APA guidelines: Use tables, but also describe what they contain Draws the reader’s attention to important features ‘Levels of reading’ journal articles and lab reports Types of statistics Descriptive statistics Summarise the data (‘shorthand’) ‘Describe the symptoms’ Inferential statistics Look at relationships within the data Draw ‘inferences’ ‘Establish the possible causes’ Indicate how ‘significant’ the results are Inferential statistics What’s the answer? What’s the question? May sound like stating the bleeding obvious, but… Establishing the specific research question to test is the hardest thing in research What is the relationship between A and B? How are A and B measured? How can differences/similarities be tested? Inferential Statistics Logic behind the test Concept, not calculation Some commonly-encountered tests Tests of Frequency Tests of Differences Tests of Association Inferential statistics How can we compare variable scores against one another? Frequency - compares frequency of membership of one category with another (nominal or ordinal) e.g. Chi-square test Differences - compares two groups in terms of a ‘score’ (interval or ratio) e.g. T-test Association - compares two variables in terms of their ‘degree of association’ (ordinal, interval or ratio) e.g. Correlation Test of frequency - Chi-square, χ 2 Compare two category variables for degree of association (non- parametric) eg: gender (male/female) vs behaviour (yes/no) Behaviour: No Yes Total: Gender: Male 60 120 180 Female 70 20 90 Total: 130 140 270 χ2 = 6.5, p=0.011 Test of frequency - Chi-square, χ2 Test statistic is “χ2” Hypothesis is “Men do it more than women” Test might be ‘significant’, but what does that mean? Compare observed frequencies with those expected by chance Behaviour: No Yes Total: Gender: Male 60Low High (90) 120 (90) 180 (45) 20Low Female 70High (45) 90 Total: 130 140 270 χ2 = 6.5, p=0.011 Test of frequency - Chi-square, χ2 More categories for ‘Yes’ response (ordinal variable) Hypothesis is “Men do it more than women” Now test might not be ‘significant’, but what does that mean? Compare observed frequencies with those expected by chance Behaviour: No A little A lot Total: Gender: Male (87) High High 60Low 60 (47) 60 (47) 180 High Low Low Female 70 (43) 10 (23) 10 (23) 90 Total: 130 70 70 270 χ2 = 4.74, p=0.093 Frequency distribution 9 Normally distributed 8 7 The area under the line 6 represents the number of 5 4 cases (respondents, 3 participants) recording 2 each score on the DV 1 0 1 2 3 4 5 6 7 Test of difference – T-test T-test Compares two groups Interval or ratio dependent variable Assumes ‘parametric’ frequency distribution “Effect of alcohol on reaction time” ‘Alcohol’ vs ‘No alcohol’ groups (IV) Reaction time (DV) is an interval variable Reaction time (DV) is normally distributed Meets the requirements of the test Hypothesis - “Alcohol will affect Reaction time” OR: “Reaction times will be slowed by alcohol” Variance ratio: the “t” statistic Broadly speaking: Area of non-overlap “t” statistic = Area of overlap Area where hypothesis true “t” statistic = Area where hypothesis false Large “t” suggests statistically ‘significant’ difference between groups This is often called the ‘variance ratio’ Hypothesis test: ‘Alcohol’ vs ‘No alcohol’ condition Hypothesis true (reaction time slower in ‘alcohol’ condition) Hypothesis false (reaction time faster in ‘alcohol’ condition) Effect of alcohol on RT Frequency No alcohol Alcohol Reaction time (ms) Test of difference – T-test Test statistic is “t” non-overlap (hypothesis true) overlap (hypothesis false) Effect Frequency No alcohol Alcohol Reaction time (ms) Test of difference – T-test Test statistic is “t” non-overlap (hypothesis true) overlap (hypothesis false) Effect Frequency No alcohol Alcohol Smaller difference between mean RT for each condition Reaction time (ms) Variance Referred to earlier as Mean R.T. (ms) Frequency measure of ‘dispersion’ Variance describes the variability in scores within a group Reaction time (ms) Top diagram shows large variance (reflected in large Frequency standard deviation) Bottom diagram shows small variance (reflected in small standard deviation) Reaction time (ms) Large dispersion Test statistic is “t” non-overlap (hypothesis true) overlap (hypothesis false) Effect Frequency No alcohol Alcohol Large standard deviation/range for groups’ reaction times Reaction time (ms) Smaller dispersion Test statistic is “t” non-overlap (hypothesis true) overlap (hypothesis false) Effect Frequency No alcohol Alcohol Smaller standard deviation/range for groups’ reaction times Reaction time (ms) Design Measures may be: Repeated (same participants in each condition) Non-repeated (different participants in each condition) Why is this important? Same participants (repeated measures) can directly compare Joe Blogg’s score in each condition can assume same shape of distribution in each condition in other words, same dispersion, different mean Different participants (non-repeated measures) must summarise differences between conditions cannot assume the same shape of distribution in each condition in other words, different dispersion in each condition Design Same participants (repeated measures) Effect Frequency Before alcohol After alcohol Reaction time (ms) Different participants (non-repeated measures) Effect Frequency Before alcohol After alcohol Reaction time (ms) Analysis of Variance (ANOVA) Test statistic is “F” non-overlap (hypothesis true) overlap (hypothesis false) Frequency 1 unit 2 units 5 units 10 units No alcohol Reaction time (ms) Variance ratio: the “F” statistic Area of non-overlap (hypothesis true) “F-ratio” = Area of overlap (hypothesis false) • Large “F” means significant differences • Large “F” means evidence in support of hypothesis • Need to calculate size of all these ‘areas’ ANOVA - important equations SS MS = Where: df MS - ‘mean square’ MSvariable SS - ‘sum of squares’ F = df - ‘degrees of freedom’ MSerror • MSvariable - the variance accounted for by the variable • MSerror - the variance not accounted for by the variable • F ratio is a variance ratio or ‘signal to noise’ ratio • Large F means large differences accounted for by the variable • Statistical significance of F ratio is reported by SPSS No alcohol 1 unit 2 units 5 units Frequency Reaction time (ms) ANOVA will tell us if one condition is significantly different to one or more of the others But it won’t tell us which conditions are different We can compare one (or more) against one (or more) of the others Need to run “contrasts”, using “contrast coefficients” Bags of flour Bag1 Bag2 Bag3 Bag4 Bag5 Five bags of flour - how heavy ? Using contrast coefficients, we can compare one against another, while ignoring the rest Which is Heavier - Bag 2 or Bag 5? Bag1 Bag2 Bag3 Bag4 Bag5 -1 +1 0 Contrast Weights are: (0, Bag 4 Bag Position: Bag1 Bag 2 Bag 3 -1, 0, 0, +1) 5 weight: 0 -1 0 0 +1 Does the average weight of 1 & 2 differ from 3 ? Bag1 Bag2 Bag3 Bag4 Bag5 Place Bag 3 at twice Answer…. What weights would you use? Why? the distance (has same Contrast weights: (?, -1, 2, 0, as 4 & 5 ?, Ignore Contrast weights: (-1, ?, ?,effect 0) placing -two ?) by 3s on normal Bag placing them scales…) at zero (pivot) -2 -1 0 +1 +2 So: (-1,-1, 0, 0) So: (?, ?, ?,2, 0, 0) so far…. Does the average weight of 1, 3 & 5 together differ from the average of 2 & 4 ? Bag 1 Bag 2 Bag 3 Bag 4 Bag 5 Answer…. What weights would you use? (?, +3, ?, +3, Contrast weights: (-2,?, ?, -2,?) -2) Why? “Polynomial” contrasts Does recall fall with delay? 15 What contrast weights should Recall performance 10 be used to answer this question? Suggestions?…. 5 Answer: (-3, -1, 1, 3) 0 1 2 3 4 …which are the weights for a linear polynomial contrast Increasing Delay Polynomial contrasts The significance of the linear contrast tells us whether the best-fitting straight line slopes downwards (or upwards!) A quadratic contrast (showing whether the best-fitting quadratic curve has a “bend” in it) would indicate whether the data plot departs from linearity (Weights for 4 levels: (1, -1, -1, 1) A cubic contrast picks up evidence for an S-shaped bend (double inflection) (Weights for 4 levels: (-1, 3, -3, 1) Weights to be used in different designs can be found on the back page of the stats booklet. Does recall fall with delay, but then recover? 18 16 What contrast weights should 14 be used to answer this question? Recall performance 12 10 Suggestions?…. 8 6 4 Answer: (1,-1,-1, 1) 2 0 …which are the weights for 0 1 2 3 4 5 a quadratic polynomial contrast Increasing Delay Does recall fall with delay, but only once a critical point is reached? 16 14 What contrast weights should 12 be used to answer this question? Recall performance 10 8 Suggestions?…. 6 Answer: (-1, 3, -3, 1) 4 2 …which are the weights for 0 a cubic polynomial contrast 0 2 4 6 8 Increasing Delay Polynomial relationships 14 16 14 12 10 Linear 12 Cubic 10 8 8 6 6 4 4 2 2 0 0 0 2 4 6 8 0 2 4 6 8 18 16 16 14 14 Quadratic 12 Quartic 12 10 10 8 8 6 6 4 4 2 2 0 0 1 2 3 4 5 0 0 2 4 6 8 10 Break Phew! I need a drink... One-way ANOVA: Driving study • Imagine an experiment on people’s estimations of vehicle distances when driving under four different circumstances: (1) Day-time driving under Clear weather conditions (2) Night driving under Clear weather conditions (3) Day-time driving under Foggy weather conditions (4) Night driving under Clear weather conditions • Suppose three drivers were tested on motorways in each case • Independent variable = driving condition Four categories or “levels” • Dependent variable = accuracy of distance estimate 1 = random guessing 10 = perfect performance One-way ANOVA: Driving study DV = “Distance estimate accuracy” IV = “Driving condition” RESULTS: Day/Clear Night/Clear Day/Foggy Night/Foggy 9 8 5 2 10 9 6 1 9 7 5 1 28 24 16 4 (Totals) 30 28 25 24 Is there a statistically 20 Accuracy 16 significant difference 15 between the conditions? 10 5 4 0 Day-clear Night-clear Day-foggy Night-foggy One-way ANOVA: Driving study Strong effect of ANOVA over the four groups gives: Driving condition - on Estimate accuracy I I I if a I • But this just means that performance in one or more of the groups was different from that in the others. • It does NOT mean that performance is affected by Day versus Night driving, or by driving in Clear versus Foggy conditions. Two-way ANOVA: Driving study 30 28 30 26 25 24 25 22 mean accuracy 20 20 Accuracy 16 15 15 14 10 10 10 5 4 5 0 0 Day-clear Night-clear Day-foggy Night-foggy Day Night Clear Foggy • Average performance in Day driving (mean = 22) is better than that for Night driving (mean = 14) • Average performance in Clear driving (mean = 26) is better than that for Foggy driving (mean = 10) • The drop in performance produced by fogginess is greater at Night (20-point drop) than it is for Day driving (12-point drop). How can we provide statistical support for these observations? Results of Two-way ANOVA on original data - II I qduifg 0 3 3 70 a I 0 n 1 0 00 1 3 7 0 3 1 3 7 0 3 1 3 7 1 3 8 0 0 2 0 1 0 a The contrast The usual suspects F-ratios Implications This means that you can use ANOVA to examine the independent effects on your data of 3, 4 or more different independent variables In the SPSS output from such analyses, the so-called main effects tell you whether each variable has an independent effect (by itself) on the dependent variable The interactions tell you how the effect of each independent variable is itself altered by the influence of the others Examples: Interaction effects 30 40 Accuracy of distance estimate Day 35 Extraverts 25 Night 30 Introverts Performance 20 25 15 Day 20 Extraverts 15 Introverts 10 10 5 Night 5 0 0 Clear Foggy Morning Afternoon • Interaction between weather • Interaction between personality conditions and time of day type and time of day • The effect of fog on distance • Extraverts bouncy in morning, judgements is greater at night slow down during course of day than during the day • Introverts take a while to get going Summary Non-parametric tests are limited in their ability to provide the experimenter with grounds for drawing conclusions - parametric tests provide more detailed information ‘Tests of difference’ use a statistic that reflects a ‘signal to noise’ ratio, or how much variance in the DV is accounted for by the IV, compared with the what is left The only fundamental difference between a t-test and ANOVA is the number of levels in the Independent Variable (IV) T-tests: IV has two levels; ANOVA: IV has three or more levels (or two or more IVs with 2+ levels) We can combine a number of IVs together in the same ANOVA procedure (two-way, three-way etc.), identifying their individual and combined (interaction) effects on the DV

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