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class: middle, inverse, title-slide # <em>t</em>-tests ### Dr Jennifer Mankin ### 21 February 2022 --- ## Looking Ahead (and Behind) - Week 4: Correlation - Last week: Chi-Square ( `\(\chi^2\)` ) -- - This week: *t*-test -- - Next week: The Linear Model - Week 8: The Linear Model --- ## Lab Report: Red Study - Today we will talk about one of the analyses for the lab report - `\(\chi^2\)` : Green study (Griskevicious et al., 2010), last week - *t*-test: Red study (Elliot et al., 2020), today! - We will talk about the lab report in the lectures **and** work on it in the practicals - Make sure you come to your registered sessions --- ## Objectives After this lecture you will understand: - The concepts behind comparing two means - Independent and paired samples *t*-tests - Where the *t*-statistic comes from - How to read histograms and means plots - How to interpret and report significance tests of *t* --- ## Comparing Two Means - Extremely common and fundamental testing paradigm - Comparing means in two groups - Independent: different entities/participants in each groups - Paired: same entities/participants in both groups - Very similar logic and interpretation, slightly different maths! --- ## Taste the Rainbow: Synaesthesia Again - People with *synaesthesia* have unusual sensory experiences - Experience colours for words, shapes for music, personalities for numbers, etc. .pull-left[  ] .pull-right[  [Image Source](https://www.discovermagazine.com/health/the-rare-humans-who-see-time-and-have-amazing-memories) ] --- ## Grapheme-Colour Synaesthesia - Association between letters/words and particular colours - Tends to be consistent throughout life, beginning in childhood - So, synaesthetes might tend to notice language/spelling more often - Research question: Do synaesthetes have a particular cognitive style, compared to non-synaesthetes? -- - Conceptual hypothesis: grapheme-colour synaesthetes have a more language-oriented cognitive style than non-synaesthetes --- ## Data and Design: SCSQ - Mealor et al (2016): Sussex Cognitive Styles Questionnaire - Includes measures of imagery, **language ability**, and more - Validated on people with and without synaesthesia -- - Operational hypothesis: Synaesthetes will, on average, have a different score on the Language subscale of the SCSQ than non-synaesthetes - Example items: "I tend to notice if a word has the same letter repeated in its spelling"; "I enjoy learning new languages" - Null hypothesis: Synaesthetes and non-synaesthetes will, on average, have the same score on the Language subscale --- ## Having a Look - All scores on the Language subscale together .codePanel[ ```r syn_data %>% ggplot(aes(x = Language)) + geom_histogram(breaks = syn_data %>% pull(Language) %>% unique(), fill = "grey") + scale_x_continuous(name = "Language and Word Forms Score", limits = c(1, 5)) + scale_y_continuous(name = "Count") ``` <!-- -->] --- ## Having a Look - Split up by grapheme-colour synaesthesia .codePanel[ ```r syn_data %>% ggplot(aes(x = Language, fill = GraphCol)) + geom_histogram(breaks = syn_data %>% pull(Language) %>% unique()) + scale_x_continuous(name = "Language and Word Forms Score", limits = c(1, 5)) + scale_y_continuous(name = "Count") + scale_fill_discrete(name = "Synaesthesia", type = c("darkcyan", "purple4")) ``` <!-- -->] --- ## Having a Look - Dotted lines for mean scores in each group .codePanel[ ```r syn_data %>% ggplot(aes(x = Language, fill = GraphCol)) + geom_histogram(breaks = syn_data %>% pull(Language) %>% unique()) + scale_x_continuous(name = "Language and Word Forms Score", limits = c(1, 5)) + scale_y_continuous(name = "Count") + scale_fill_discrete(name = "Synaesthesia", type = c("darkcyan", "purple4")) + geom_vline(aes(xintercept = mean(Language)), data = syn_data %>% filter(GraphCol == "Yes"), colour = "purple3", linetype = "dashed") + geom_vline(aes(xintercept = mean(Language)), data = syn_data %>% filter(GraphCol == "No"), colour = "turquoise3", linetype = "dashed") ``` <!-- -->] --- ## Sorted! - The mean Language score for synaesthetes is higher than for non-synaesthetes - Are we done? -- - Of course not 😉 - How different are these mean scores, *accounting for the variation in scores*? - How strong is the *signal* (the difference in means)... - Compared to the *noise* (the variation in scores around the mean)? --- ## Having a Closer Look - Plot of the means in each group - Error bars represent 2 `\(\times\)` the standard error - How can we interpret this plot? .codePanel[ ```r syn_data %>% dplyr::group_by(GraphCol) %>% dplyr::summarise( n = dplyr::n(), mean_lang = mean(Language), se_lang = sd(Language)/sqrt(n) ) %>% ggplot2::ggplot(aes(x = GraphCol, y = mean_lang)) + geom_errorbar(aes(ymin = mean_lang - 2*se_lang, ymax = mean_lang + 2*se_lang), width = .1) + geom_point(colour = "black", fill = "orange", pch = 23) + scale_y_continuous(name = "Language Score", limits = c(1, 5)) + labs(x = "Grapheme-Colour Synaesthete") + cowplot::theme_cowplot() ``` <!-- -->] --- ## Why You Gotta Be So Mean - Let's get some numbers to work with 😁 - Mean in synaesthete group: 4.29 - Mean in nonsynaesthete group: 3.55 -- - Difference in the means = 4.29 - 3.55 = 0.74 -- - Is this a big difference, compared to how different we might expect for any two sample means **from the same population**? --- ## Steps of the Analysis - Calculate the (standardised) *difference* between mean scores -- - Compare that *test statistic* to its distribution under the null hypothesis -- - Obtain the probability *p* of encountering a test statistic of the size we have, or larger, if the null hypothesis is true -- ``` ## ## Two Sample t-test ## ## data: Language by GraphCol ## t = -6.3394, df = 1209, p-value = 0.000000000325 ## alternative hypothesis: true difference in means between group No and group Yes is not equal to 0 ## 95 percent confidence interval: ## -0.9576637 -0.5049972 ## sample estimates: ## mean in group No mean in group Yes ## 3.554949 4.286279 ``` - Where does this number *t* come from? And what does it mean? --- ## Let's Think About This - Imagine we took two samples of scores from a single population and calculated their means, and then found the **difference** between those means - Create a sampling distribution of *mean differences* - Centre of the distribution = 0 (no difference between the means) - Very small differences in means will be quite common - Very large differences in means will be quite unlikely -- - Distribution under the **null hypothesis** that which group you're in (e.g. synaesthete vs non-synaesthete) makes no difference to your score - Samples come from the same population of scores --- ## Sound Familiar? - If this rings a bell, it should! - This is the **same scenario** from [Lecture 3](https://and.netlify.app/lectures/03/slides/#42) for the Ape Index (AI) <img class="gif bottom" src="/lectures_assets/05/no_diff1.png" gif="/lectures_assets/05/no_diff2.png /lectures_assets/05/no_diff_1_small.gif" height="280px"> --- ## Sound Familiar? - Our estimate of mean Language score difference (0.74) also has some distribution - Using that distribution, obtain the probability *p* of finding a difference as large as the one we found, or larger, if the null hypothesis is true (as usual) -- - **Standardise** our mean difference to compare it to a known distribution - We can do this by dividing the mean difference by the standard error - In other words: *scale* it to the distribution - Conceptually similar to [*z*-scores](https://paas.netlify.app/lectures/lec08/handout/#the-z-transform) --- ## Same But Different - For the AI in Lecture 3, the standard deviation of the population was 1 - Assumed for simplicity - Remember, `\(SE = \frac{\sigma}{\sqrt{n}}\)` -- - For our synaesthesia example, the standard error is a **variable** - We estimate `\(\hat{SE}\)` based on the standard deviation *s* - Every sample, even samples of the same size, will have a different *s*! --- ## What's the Poin*t*? - The test statistic *t* is the difference in group means divided by the standard error - Difference in group means: the effect of interest, or *signal* - Standard error of the difference in means: the error, or *noise* - So, *t* represents how big the signal is compared to the noise - Larger values of *t* are more unlikely under the null hypothesis (when the "signal" is really 0!) --- ## Would You Like Some *t*? - Naturally, *t* is *t*-distributed (here, with *N*<sub>1</sub> + *N*<sub>2</sub> - 2 degrees of freedom) - Given an `\(\alpha\)` level of .05... - If *p* > .05, we conclude that our results are *likely* to occur under the null hypothesis, so we have no evidence that the null hypothesis is not true - If *p* < .05, we conclude that our results are sufficiently *unlikely* to occur that it may in fact be the case that the null hypothesis is *not* true <iframe id="t-app" class="app" src="https://and.netlify.app/viz/tdist.html" height=680px></iframe> --- ## Would You Like Some *t*? ``` ## ## Two Sample t-test ## ## data: Language by GraphCol ## t = -6.3394, df = 1209, p-value = 0.000000000325 ## alternative hypothesis: true difference in means between group No and group Yes is not equal to 0 ## 95 percent confidence interval: ## -0.9576637 -0.5049972 ## sample estimates: ## mean in group No mean in group Yes ## 3.554949 4.286279 ``` <br> .center[ "On average, grapheme-colour synaesthetes scored higher on the Language subscale of the SCSQ (*M* = 4.29, *SD* = 0.7) than non-synaesthetes (*M* = 3.55, *SD* = 0.74). An independent samples *t*-test revealed that this difference was statistically significant (*t*(1209) = -6.34, *p* < .001, *M*<sub>diff</sub> = -0.74, 95% CI [-0.96, -0.5])." ] --- exclude: ![:live] .pollEv[ <iframe src="https://embed.polleverywhere.com/discourses/IXL29dEI4tgTYG0zH1Sef?controls=none&short_poll=true" width="800px" height="600px"></iframe> ] --- ## Interim Summary - Independent samples *t*-test - Tests whether two different samples come from the same population using *t* - If *p* < `\(\alpha\)` (typically .05), then it is *unlikely* that they do - So, we conclude that group membership is associated with some difference - If you choose the Red study, **this is the test you will use** -- - Next up: paired samples *t*-test --- ## Do You Want Some Synaesthesia? - Being a synaesthete is super cool and a lot of fun - See cool colours all the time! - Have (very mundane and mostly unremarkable) superpowers! -- - What if everyone could be a synaesthete? - Can you **train** people to have synaesthesia? --- ## Paired (Repeated) Design - Simplified version of [Bor et al. (2014)](https://www.nature.com/articles/srep07089#Sec22) - Train people to associate colours with letters - Test success of the training with a modified Stroop task - Outcome: naming speed pre- vs post-training  --- ## Paired (Repeated) Design - Key difference: the **same people** participate in both conditions .pull-left[ ``` ## # A tibble: 6 x 2 ## pre post ## <dbl> <dbl> ## 1 1092. 1034. ## 2 765. 709. ## 3 970. 911. ## 4 836. 776. ## 5 778. 716. ## 6 871. 810. ``` ] .pull-right[ - The data are *paired* - Each row contains the same thing (here, reaction time) - Each row contains data from the same person ] --- ## Example Output ``` ## ## Paired t-test ## ## data: syn_train$pre and syn_train$post ## t = 96.493, df = 13, p-value < 0.00000000000000022 ## alternative hypothesis: true difference in means is not equal to 0 ## 95 percent confidence interval: ## 57.53939 60.17489 ## sample estimates: ## mean of the differences ## 58.85714 ``` .center[ "There was a significant difference in mean colour naming times between pre- and post-training (*t*(13) = 96.49, *p* < .001, *M*<sub>diff</sub> = 58.86, 95% CI [57.54, 60.17])." ] --- ## Causality - In our first example, could we conclude that having synaesthesia *causes* you to pay more attention to language? - In our second example, could we conclude that having training *causes* you to associate colours with letters? -- - Why is this? --- ## That's the *t* - The *t*-test quantifies the size of the difference of two means (signal) compared to the error (noise) - Independent samples *t*-test - Tests means from different entities/participants - Independent or "between-subjects" design - Paired samples *t*-test - Tests means from the same entities/participants - Repeated or "within-subjects" design - Establishing causality is a function of **study design** not statistics! --- ## Lab Reports - You can choose either the red or green study to write your report on - See [Lab Report Information on Canvas](https://canvas.sussex.ac.uk/courses/9242/pages/Lab%20Report%20Information%20and%20Resources?titleize=0) for more - If you choose the red study (Elliot et al., 2010), you must use and report the results of an independent samples *t*-test -- - Create a composite score out of the three rating scales - Report means, SDs, and *t*-test result - Include a figure of the results - Will be covered in depth in the next tutorial and practical! --- exclude: ![:live] .pollEv[ <iframe src="https://embed.polleverywhere.com/discourses/IXL29dEI4tgTYG0zH1Sef?controls=none&short_poll=true" width="800px" height="600px"></iframe> ] --- class: last-slide
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