Some Kind of End to the Course

Univariate Statistics and Methodology using R

Martin Corley

Psychology, PPLS

University of Edinburgh

Zen and the Art of Stats

Null-Hypothesis Statistical Testing

founded on two principles

you can’t prove a hypothesis to be true

the sun will rise every day

just one counterexample…

Null-Hypothesis Statistical Testing

founded on two principles

you can’t prove a hypothesis to be true
the universe is governed by chance

Null-Hypothesis Statistical Testing

founded on two principles

you can’t prove a hypothesis to be true
the universe is governed by chance

A Theory about the Structure of Chance

if we deviate enough from that structure, we assume that something else is at play

What is Enough?

enough is operationalised as a probability
- based on summary statistics (e.g., \(\bar{x}, \sigma, n, \textrm{se}\))
summaries assume that chance is ruling things
summary statistics are often
- \(\frac{x-\bar{x}}{\sigma}\) (single observations)
- \(\frac{\bar{x}}{\textrm{se}}\) (group summaries)
(these are basically the same thing, compared to \(z\) or \(t\) as appropriate)

if the probability of obtaining the data we have observed (or more extreme data) under the null hypothesis is low enough…

we have made the right assumptions about chance
we have calculated the right summary
we have looked up the appropriate probability (of obtaining summary statistic or more)
that probability is below the \(\alpha\) level we have previously set

then

we can, tentatively, suggest that maybe the pattern we have observed is not due to chance

But Linear Models…?

\[\color{red}{\textrm{outcome}_i} = \color{blue}{(\textrm{model})_i} + \textrm{error}_i\]

essentially, these are based on covariance
the model expresses the amount that variables covary
the error is the unexplained variance, and we have a theory about its distribution
we want the model (SS) to be big compared to the error (SS)
- the \(F\) ratio measures this (so does \(R^2\), less directly)
again, we’re calculating summary statistics and looking up the probability of obtaining them (or more) in a chance universe

Coefficients

coefficients tell you about (hypothetical) lines through data
the most important (and often least exciting) coefficient is the intercept
- other properties of the line are always relative to this
- we can change the interpretation of the intercept by scaling predictors

when predictors are categories

we assign numbers to the category values (one pair of values for each of \(n-1\) comparisons)
- the interpretation of the intercept depends on these values

Non-Linear Outcomes

use generalized linear models
specify link function
most frequent case is binomial

→ logit (log-odds) link function

Non-Linear Outcomes

main things to remember:
- we’re dealing in deviance, not variance
- how to convert the coefficients

→ p/(1-p)

← o/(o+1)

→ log(o)

← exp(l)

some papers talk about odds, some about probabilities

Model Building

When Should I Put Things in My Models?

to estimate and test an association/difference of interest
to get the “best” fitting/predicting model

psych research typically interested in (1).

Control Isn’t Always Easy

draw your theory out:

model <- lm(grade ~ attendance)

Control Isn’t Always Easy

draw your theory out:

model <- lm(grade ~ knowledge + attendance)

Control Isn’t Always Easy

draw your theory out:

maybe “knowledge” isn’t a good control variable?

Conclusions Aren’t Always Easy

draw your theory out:

Conclusions Aren’t Always Easy

Matthews (2000). https://doi.org/10.1111/1467-9639.00013

Conclusions Aren’t Always Easy

the direction is something we reason about
the model doesn’t care:

lm (y ~ x) and lm (x ~ y) are “the same” (cor.test(y,x))

Correlation doesn’t imply causation, but it does waggle its eyebrows suggestively and gesture furtively while mouthing ‘look over there’.

Randall Munroe https://xkcd.com/552

The Bits We Didn’t Talk About

ANOVA

If you should say to a mathematical statistician that you have discovered that linear multiple regression and the analysis of variance (and covariance) are identical systems, he would mutter something like “Of course—general linear model,” and you might have trouble maintaining his attention. If you should say this to a typical psychologist, you would be met with incredulity, or worse. Yet it is true, and in its truth lie possibilities for more relevant and therefore more powerful research data.

Cohen, 1968

History

Multiple Regression

introduced c. 1900 in biological and behavioural sciences
aligned to “natural variation” in observations
shows that means \(\bar{y}\) are related to groups \(g_1,g_2,\ldots,g_n\)

ANOVA

introduced c. 1920 in agricultural research
aligned to experimentation and manipulation
shows that groups \(g_1,g_2,\ldots,g_n\) have different means \(\bar{y}\)

both produce \(F\)-ratios, discussed in different language, but identical

Why Teach GLM/Regression?

GLM has less restrictive assumptions
- especially true for unbalanced designs/missing data
GLM is far better at dealing with covariates
- can arbitrarily mix continuous and discrete predictors
GLM is the gateway to other powerful tools
- mixed models and factor analysis (→ MSMR)
- structural equation models

ANOVA in R

type	UTILITY
zing	4.1
playmo	7.3
lego	4.6
zing	2.2
playmo	7.9
lego	5.1
zing	3.9
playmo	6.6
lego	5.7
zing	1.2

GLM vs ANOVA

l.mod <- lm(UTILITY ~ type, data = toys)
anova(l.mod)

Analysis of Variance Table

Response: UTILITY
          Df Sum Sq Mean Sq F value   Pr(>F)    
type       2   83.3    41.7    25.1 0.000052 ***
Residuals 12   19.9     1.7                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

a.mod <- aov(UTILITY ~ type, data = toys)
summary(a.mod)

            Df Sum Sq Mean Sq F value   Pr(>F)    
type         2   83.3    41.7    25.1 0.000052 ***
Residuals   12   19.9     1.7                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

GLM vs ANOVA

summary(l.mod)  #<<

...
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    2.280      0.577    3.95   0.0019 ** 
typeplaymo     5.760      0.815    7.06 0.000013 ***
typelego       2.540      0.815    3.11   0.0089 ** 
...

model.tables(a.mod)

Tables of effects

 type 
type
   zing  playmo    lego 
-2.7667  2.9933 -0.2267

The Elephant in the Room

Repeated Measures

so far, every model we’ve looked at has been “one observation per participant”
however, most experiments have a structure
some observations are “more related” to each other than others
for example, because they come from the same person (repeated measures)

Mixed-Effects Models

\[y_{ij}=b_{0j}+b_{1j}x_{1ij}+\epsilon_{ij}\]

\[b_{0j}=\gamma_{00}+\zeta_{0j}\] \[b_{1j}=\gamma_{01}+\zeta_{1j}\]

relatedness accounted for by more regression equations
all part of the linear model → next semester

Reaching Nirv-R-na

R is a Language

no ‘perfect way’ to speak it
- ‘dialects’ such as {tidyverse} and {data.table}
constantly changing
- new libraries ({magrittr} introduced %>%)
- new syntax (R v4.1.0 introduced |>)
we are all constantly learning new things

Learn by Taking It Apart

singers |>
    mutate(bin = cut_interval(quality,
        10)) |>
    group_by(bin) |>
    summarise(prop = mean(SPLATTED)) |>
    ggplot(aes(x = bin, y = prop)) +
    xlab("quality bin") +
    ylab("prop splatted") +
    geom_point(size = 3) +
    scale_x_discrete(label = 1:10)

Learn by Taking It Apart

singers |>
  mutate(bin = cut_interval(quality, 10))

        id quality SPLATTED         bin
1    A0001      84        0 (80.2,90.1]
2    A0002      34        1 (30.7,40.6]
3    A0003      92        0  (90.1,100]
4    A0004      49        1 (40.6,50.5]
5    A0005      93        0  (90.1,100]
6    A0006       5        1    [1,10.9]
7    A0007      55        1 (50.5,60.4]
8    A0008      64        1 (60.4,70.3]
9    A0009      45        0 (40.6,50.5]
10   A0010      90        0 (80.2,90.1]
11   A0011      51        1 (50.5,60.4]
12   A0012       1        1    [1,10.9]
13   A0013      43        0 (40.6,50.5]
14   A0014      89        0 (80.2,90.1]
15   A0015      72        0 (70.3,80.2]
16   A0016      46        1 (40.6,50.5]
17   A0017      20        1 (10.9,20.8]
18   A0018      68        0 (60.4,70.3]
19   A0019      10        1    [1,10.9]
20   A0020      75        1 (70.3,80.2]
21   A0021      18        1 (10.9,20.8]
22   A0022      44        0 (40.6,50.5]
23   A0023      43        0 (40.6,50.5]
24   A0024      61        0 (60.4,70.3]
25   A0025      57        1 (50.5,60.4]
26   A0026      73        0 (70.3,80.2]
27   A0027      95        0  (90.1,100]
28   A0028      78        0 (70.3,80.2]
29   A0029      53        0 (50.5,60.4]
30   A0030      22        1 (20.8,30.7]
31   A0031      75        0 (70.3,80.2]
32   A0032      49        1 (40.6,50.5]
33   A0033      21        1 (20.8,30.7]
34   A0034      85        0 (80.2,90.1]
35   A0035      60        0 (50.5,60.4]
36   A0036      91        0  (90.1,100]
37   A0037      89        0 (80.2,90.1]
38   A0038      71        0 (70.3,80.2]
39   A0039      93        0  (90.1,100]
40   A0040      52        1 (50.5,60.4]
41   A0041       5        1    [1,10.9]
42   A0042      32        1 (30.7,40.6]
43   A0043      84        0 (80.2,90.1]
44   A0044      55        0 (50.5,60.4]
45   A0045      26        1 (20.8,30.7]
46   A0046      99        0  (90.1,100]
47   A0047      78        0 (70.3,80.2]
48   A0048      68        0 (60.4,70.3]
49   A0049      64        0 (60.4,70.3]
50   A0050      57        1 (50.5,60.4]
51   A0051      63        0 (60.4,70.3]
52   A0052      60        0 (50.5,60.4]
53   A0053      37        0 (30.7,40.6]
54   A0054      82        0 (80.2,90.1]
55   A0055      66        0 (60.4,70.3]
56   A0056      40        1 (30.7,40.6]
57   A0057       7        1    [1,10.9]
58   A0058      10        1    [1,10.9]
59   A0059      45        0 (40.6,50.5]
60   A0060      84        0 (80.2,90.1]
61   A0061      74        0 (70.3,80.2]
62   A0062      23        1 (20.8,30.7]
63   A0063     100        0  (90.1,100]
64   A0064      40        1 (30.7,40.6]
65   A0065      10        1    [1,10.9]
66   A0066      77        0 (70.3,80.2]
67   A0067      28        1 (20.8,30.7]
68   A0068       3        1    [1,10.9]
69   A0069      84        0 (80.2,90.1]
70   A0070      37        1 (30.7,40.6]
71   A0071      31        1 (30.7,40.6]
72   A0072       4        1    [1,10.9]
73   A0073      37        1 (30.7,40.6]
74   A0074      31        1 (30.7,40.6]
75   A0075      13        1 (10.9,20.8]
76   A0076      71        0 (70.3,80.2]
77   A0077      85        0 (80.2,90.1]
78   A0078      94        0  (90.1,100]
79   A0079      48        0 (40.6,50.5]
80   A0080      94        0  (90.1,100]
81   A0081      10        1    [1,10.9]
82   A0082       1        1    [1,10.9]
83   A0083      93        0  (90.1,100]
84   A0084      66        0 (60.4,70.3]
85   A0085      93        0  (90.1,100]
86   A0086      24        1 (20.8,30.7]
87   A0087       9        1    [1,10.9]
88   A0088      79        0 (70.3,80.2]
89   A0089      13        1 (10.9,20.8]
90   A0090      97        0  (90.1,100]
91   A0091      85        0 (80.2,90.1]
92   A0092      35        1 (30.7,40.6]
93   A0093      28        1 (20.8,30.7]
94   A0094      79        0 (70.3,80.2]
95   A0095      51        0 (50.5,60.4]
96   A0096      85        0 (80.2,90.1]
97   A0097      97        0  (90.1,100]
98   A0098      45        0 (40.6,50.5]
99   A0099      49        1 (40.6,50.5]
100  A0100      34        1 (30.7,40.6]
101  A0101      19        1 (10.9,20.8]
102  A0102      22        1 (20.8,30.7]
103  A0103       3        1    [1,10.9]
104  A0104      11        1 (10.9,20.8]
105  A0105      42        0 (40.6,50.5]
106  A0106      58        0 (50.5,60.4]
107  A0107      63        0 (60.4,70.3]
108  A0108      37        1 (30.7,40.6]
109  A0109      68        0 (60.4,70.3]
110  A0110      70        0 (60.4,70.3]
111  A0111      85        0 (80.2,90.1]
112  A0112       3        1    [1,10.9]
113  A0113      47        1 (40.6,50.5]
114  A0114      20        1 (10.9,20.8]
115  A0115      85        0 (80.2,90.1]
116  A0116      98        0  (90.1,100]
117  A0117      42        1 (40.6,50.5]
118  A0118      60        0 (50.5,60.4]
119  A0119      70        0 (60.4,70.3]
120  A0120      26        1 (20.8,30.7]
121  A0121      94        0  (90.1,100]
122  A0122      30        1 (20.8,30.7]
123  A0123       5        1    [1,10.9]
124  A0124      93        1  (90.1,100]
125  A0125       9        1    [1,10.9]
126  A0126      64        0 (60.4,70.3]
127  A0127      74        0 (70.3,80.2]
128  A0128      75        0 (70.3,80.2]
129  A0129      81        0 (80.2,90.1]
130  A0130      70        0 (60.4,70.3]
131  A0131      35        1 (30.7,40.6]
132  A0132      12        1 (10.9,20.8]
133  A0133      85        0 (80.2,90.1]
134  A0134      66        0 (60.4,70.3]
135  A0135      62        0 (60.4,70.3]
136  A0136      98        0  (90.1,100]
137  A0137      61        0 (60.4,70.3]
138  A0138      27        1 (20.8,30.7]
139  A0139       4        1    [1,10.9]
140  A0140      30        1 (20.8,30.7]
141  A0141      49        0 (40.6,50.5]
142  A0142      89        0 (80.2,90.1]
143  A0143      97        0  (90.1,100]
144  A0144      82        0 (80.2,90.1]
145  A0145      12        1 (10.9,20.8]
146  A0146      96        0  (90.1,100]
147  A0147      97        0  (90.1,100]
148  A0148      27        1 (20.8,30.7]
149  A0149      86        0 (80.2,90.1]
150  A0150      14        1 (10.9,20.8]
151  A0151      76        0 (70.3,80.2]
152  A0152      80        0 (70.3,80.2]
153  A0153      12        1 (10.9,20.8]
154  A0154      97        0  (90.1,100]
155  A0155      53        1 (50.5,60.4]
156  A0156      55        0 (50.5,60.4]
157  A0157      76        0 (70.3,80.2]
158  A0158      85        0 (80.2,90.1]
159  A0159      16        1 (10.9,20.8]
160  A0160     100        0  (90.1,100]
161  A0161      20        1 (10.9,20.8]
162  A0162      64        1 (60.4,70.3]
163  A0163      14        1 (10.9,20.8]
164  A0164      44        1 (40.6,50.5]
165  A0165      93        0  (90.1,100]
166  A0166      35        1 (30.7,40.6]
167  A0167      44        0 (40.6,50.5]
168  A0168      46        1 (40.6,50.5]
169  A0169      67        0 (60.4,70.3]
170  A0170      81        0 (80.2,90.1]
171  A0171      88        0 (80.2,90.1]
172  A0172       6        1    [1,10.9]
173  A0173      26        1 (20.8,30.7]
174  A0174      89        0 (80.2,90.1]
175  A0175       8        1    [1,10.9]
176  A0176      88        0 (80.2,90.1]
177  A0177      29        1 (20.8,30.7]
178  A0178      49        0 (40.6,50.5]
179  A0179      91        0  (90.1,100]
180  A0180      60        0 (50.5,60.4]
181  A0181      74        0 (70.3,80.2]
182  A0182      10        1    [1,10.9]
183  A0183      52        1 (50.5,60.4]
184  A0184      74        0 (70.3,80.2]
185  A0185      69        0 (60.4,70.3]
186  A0186      74        0 (70.3,80.2]
187  A0187      31        1 (30.7,40.6]
188  A0188      95        0  (90.1,100]
189  A0189      86        0 (80.2,90.1]
190  A0190      28        1 (20.8,30.7]
191  A0191      31        1 (30.7,40.6]
192  A0192      72        0 (70.3,80.2]
193  A0193      80        0 (70.3,80.2]
194  A0194      88        0 (80.2,90.1]
195  A0195      65        0 (60.4,70.3]
196  A0196      61        0 (60.4,70.3]
197  A0197      46        0 (40.6,50.5]
198  A0198       7        1    [1,10.9]
199  A0199      93        0  (90.1,100]
200  A0200      87        0 (80.2,90.1]
201  A0201       6        1    [1,10.9]
202  A0202      56        0 (50.5,60.4]
203  A0203      79        0 (70.3,80.2]
204  A0204      71        0 (70.3,80.2]
205  A0205      10        1    [1,10.9]
206  A0206      63        0 (60.4,70.3]
207  A0207       2        1    [1,10.9]
208  A0208      99        0  (90.1,100]
209  A0209      45        1 (40.6,50.5]
210  A0210      81        0 (80.2,90.1]
211  A0211      23        1 (20.8,30.7]
212  A0212      66        0 (60.4,70.3]
213  A0213      10        1    [1,10.9]
214  A0214      10        1    [1,10.9]
215  A0215      99        1  (90.1,100]
216  A0216      76        0 (70.3,80.2]
217  A0217      33        1 (30.7,40.6]
218  A0218      48        1 (40.6,50.5]
219  A0219      10        1    [1,10.9]
220  A0220      73        1 (70.3,80.2]
221  A0221      20        1 (10.9,20.8]
222  A0222      62        1 (60.4,70.3]
223  A0223      27        1 (20.8,30.7]
224  A0224      29        1 (20.8,30.7]
225  A0225      31        1 (30.7,40.6]
226  A0226      50        1 (40.6,50.5]
227  A0227      29        1 (20.8,30.7]
228  A0228      91        0  (90.1,100]
229  A0229      53        0 (50.5,60.4]
230  A0230      84        0 (80.2,90.1]
231  A0231      50        1 (40.6,50.5]
232  A0232      93        0  (90.1,100]
233  A0233      69        0 (60.4,70.3]
234  A0234      84        0 (80.2,90.1]
235  A0235      35        1 (30.7,40.6]
236  A0236      58        0 (50.5,60.4]
237  A0237      48        1 (40.6,50.5]
238  A0238      16        1 (10.9,20.8]
239  A0239      72        0 (70.3,80.2]
240  A0240      64        0 (60.4,70.3]
241  A0241      62        0 (60.4,70.3]
242  A0242      47        0 (40.6,50.5]
243  A0243      21        1 (20.8,30.7]
244  A0244      72        0 (70.3,80.2]
245  A0245      80        0 (70.3,80.2]
246  A0246      76        0 (70.3,80.2]
247  A0247      39        0 (30.7,40.6]
248  A0248     100        0  (90.1,100]
249  A0249      44        0 (40.6,50.5]
250  A0250      59        1 (50.5,60.4]
251  A0251      38        0 (30.7,40.6]
252  A0252      14        1 (10.9,20.8]
253  A0253      73        0 (70.3,80.2]
254  A0254      87        0 (80.2,90.1]
255  A0255      28        1 (20.8,30.7]
256  A0256      46        0 (40.6,50.5]
257  A0257      51        1 (50.5,60.4]
258  A0258       7        1    [1,10.9]
259  A0259      63        0 (60.4,70.3]
260  A0260      25        1 (20.8,30.7]
261  A0261      79        0 (70.3,80.2]
262  A0262      47        1 (40.6,50.5]
263  A0263      77        0 (70.3,80.2]
264  A0264      85        0 (80.2,90.1]
265  A0265      30        1 (20.8,30.7]
266  A0266      20        1 (10.9,20.8]
267  A0267      87        0 (80.2,90.1]
268  A0268      18        1 (10.9,20.8]
269  A0269      96        0  (90.1,100]
270  A0270      21        1 (20.8,30.7]
271  A0271      98        0  (90.1,100]
272  A0272      40        0 (30.7,40.6]
273  A0273      69        0 (60.4,70.3]
274  A0274      14        1 (10.9,20.8]
275  A0275      28        1 (20.8,30.7]
276  A0276      74        0 (70.3,80.2]
277  A0277      41        1 (40.6,50.5]
278  A0278      72        0 (70.3,80.2]
279  A0279       7        1    [1,10.9]
280  A0280      86        0 (80.2,90.1]
281  A0281      41        1 (40.6,50.5]
282  A0282      72        0 (70.3,80.2]
283  A0283      58        0 (50.5,60.4]
284  A0284      98        0  (90.1,100]
285  A0285      70        0 (60.4,70.3]
286  A0286      18        1 (10.9,20.8]
287  A0287      51        1 (50.5,60.4]
288  A0288      17        1 (10.9,20.8]
289  A0289      66        0 (60.4,70.3]
290  A0290      59        0 (50.5,60.4]
291  A0291      10        1    [1,10.9]
292  A0292      57        0 (50.5,60.4]
293  A0293      80        0 (70.3,80.2]
294  A0294      88        0 (80.2,90.1]
295  A0295      89        0 (80.2,90.1]
296  A0296      27        1 (20.8,30.7]
297  A0297      92        0  (90.1,100]
298  A0298      71        0 (70.3,80.2]
299  A0299      96        0  (90.1,100]
300  A0300      78        0 (70.3,80.2]
301  A0301      64        0 (60.4,70.3]
302  A0302      72        0 (70.3,80.2]
303  A0303      70        0 (60.4,70.3]
304  A0304      82        0 (80.2,90.1]
305  A0305       9        1    [1,10.9]
306  A0306      76        0 (70.3,80.2]
307  A0307       5        1    [1,10.9]
308  A0308      70        0 (60.4,70.3]
309  A0309       8        1    [1,10.9]
310  A0310      78        0 (70.3,80.2]
311  A0311       6        1    [1,10.9]
312  A0312      98        0  (90.1,100]
313  A0313      83        0 (80.2,90.1]
314  A0314      65        0 (60.4,70.3]
315  A0315      48        0 (40.6,50.5]
316  A0316      29        1 (20.8,30.7]
317  A0317      25        1 (20.8,30.7]
318  A0318      97        0  (90.1,100]
319  A0319      19        0 (10.9,20.8]
320  A0320      49        0 (40.6,50.5]
321  A0321      48        0 (40.6,50.5]
322  A0322      58        1 (50.5,60.4]
323  A0323      39        0 (30.7,40.6]
324  A0324      57        0 (50.5,60.4]
325  A0325      84        0 (80.2,90.1]
326  A0326      24        1 (20.8,30.7]
327  A0327      44        1 (40.6,50.5]
328  A0328      69        0 (60.4,70.3]
329  A0329      47        0 (40.6,50.5]
330  A0330      65        0 (60.4,70.3]
331  A0331      79        0 (70.3,80.2]
332  A0332      11        1 (10.9,20.8]
333  A0333      82        0 (80.2,90.1]
334  A0334      90        0 (80.2,90.1]
335  A0335      67        0 (60.4,70.3]
336  A0336      53        1 (50.5,60.4]
337  A0337      81        0 (80.2,90.1]
338  A0338      90        0 (80.2,90.1]
339  A0339      41        1 (40.6,50.5]
340  A0340       3        1    [1,10.9]
341  A0341      17        1 (10.9,20.8]
342  A0342      55        0 (50.5,60.4]
343  A0343      22        1 (20.8,30.7]
344  A0344      76        0 (70.3,80.2]
345  A0345      10        1    [1,10.9]
346  A0346      61        0 (60.4,70.3]
347  A0347      61        0 (60.4,70.3]
348  A0348      97        0  (90.1,100]
349  A0349       8        1    [1,10.9]
350  A0350      77        0 (70.3,80.2]
351  A0351      48        0 (40.6,50.5]
352  A0352      16        1 (10.9,20.8]
353  A0353      80        0 (70.3,80.2]
354  A0354      26        0 (20.8,30.7]
355  A0355      37        1 (30.7,40.6]
356  A0356      45        0 (40.6,50.5]
357  A0357      21        1 (20.8,30.7]
358  A0358      30        1 (20.8,30.7]
359  A0359       7        1    [1,10.9]
360  A0360       6        1    [1,10.9]
361  A0361      76        0 (70.3,80.2]
362  A0362      44        1 (40.6,50.5]
363  A0363      57        0 (50.5,60.4]
364  A0364      43        0 (40.6,50.5]
365  A0365      46        1 (40.6,50.5]
366  A0366      59        0 (50.5,60.4]
367  A0367      75        0 (70.3,80.2]
368  A0368      73        0 (70.3,80.2]
369  A0369      62        0 (60.4,70.3]
370  A0370      75        0 (70.3,80.2]
371  A0371      72        0 (70.3,80.2]
372  A0372      13        1 (10.9,20.8]
373  A0373      40        0 (30.7,40.6]
374  A0374      46        0 (40.6,50.5]
375  A0375       2        1    [1,10.9]
376  A0376      95        0  (90.1,100]
377  A0377      78        0 (70.3,80.2]
378  A0378      67        0 (60.4,70.3]
379  A0379      16        1 (10.9,20.8]
380  A0380      95        0  (90.1,100]
381  A0381      38        1 (30.7,40.6]
382  A0382      69        0 (60.4,70.3]
383  A0383      88        0 (80.2,90.1]
384  A0384      87        0 (80.2,90.1]
385  A0385      95        0  (90.1,100]
386  A0386       8        1    [1,10.9]
387  A0387      28        1 (20.8,30.7]
388  A0388       8        1    [1,10.9]
389  A0389      95        1  (90.1,100]
390  A0390       8        1    [1,10.9]
391  A0391      60        0 (50.5,60.4]
392  A0392      39        0 (30.7,40.6]
393  A0393      38        1 (30.7,40.6]
394  A0394      44        1 (40.6,50.5]
395  A0395      51        1 (50.5,60.4]
396  A0396      95        1  (90.1,100]
397  A0397      82        0 (80.2,90.1]
398  A0398      17        1 (10.9,20.8]
399  A0399      55        0 (50.5,60.4]
400  A0400      96        0  (90.1,100]
401  A0401      44        1 (40.6,50.5]
402  A0402      36        1 (30.7,40.6]
403  A0403      23        1 (20.8,30.7]
404  A0404      14        1 (10.9,20.8]
405  A0405      91        0  (90.1,100]
406  A0406      87        0 (80.2,90.1]
407  A0407       1        1    [1,10.9]
408  A0408      97        0  (90.1,100]
409  A0409      39        0 (30.7,40.6]
410  A0410      92        0  (90.1,100]
411  A0411      52        0 (50.5,60.4]
412  A0412       9        1    [1,10.9]
413  A0413      31        1 (30.7,40.6]
414  A0414      89        0 (80.2,90.1]
415  A0415      59        0 (50.5,60.4]
416  A0416      63        1 (60.4,70.3]
417  A0417      91        0  (90.1,100]
418  A0418      90        0 (80.2,90.1]
419  A0419      69        0 (60.4,70.3]
420  A0420      62        0 (60.4,70.3]
421  A0421      60        0 (50.5,60.4]
422  A0422      57        1 (50.5,60.4]
423  A0423      51        0 (50.5,60.4]
424  A0424      90        0 (80.2,90.1]
425  A0425      95        0  (90.1,100]
426  A0426      15        1 (10.9,20.8]
427  A0427       9        1    [1,10.9]
428  A0428      24        1 (20.8,30.7]
429  A0429      18        1 (10.9,20.8]
430  A0430      95        0  (90.1,100]
431  A0431      11        1 (10.9,20.8]
432  A0432      67        0 (60.4,70.3]
433  A0433      57        0 (50.5,60.4]
434  A0434      11        1 (10.9,20.8]
435  A0435      72        0 (70.3,80.2]
436  A0436      77        0 (70.3,80.2]
437  A0437      74        0 (70.3,80.2]
438  A0438      19        1 (10.9,20.8]
439  A0439      59        1 (50.5,60.4]
440  A0440      51        1 (50.5,60.4]
441  A0441      56        0 (50.5,60.4]
442  A0442      67        0 (60.4,70.3]
443  A0443      36        1 (30.7,40.6]
444  A0444      55        0 (50.5,60.4]
445  A0445      98        0  (90.1,100]
446  A0446      93        0  (90.1,100]
447  A0447      37        1 (30.7,40.6]
448  A0448      77        0 (70.3,80.2]
449  A0449      90        0 (80.2,90.1]
450  A0450      88        0 (80.2,90.1]
451  A0451       7        1    [1,10.9]
452  A0452      35        1 (30.7,40.6]
453  A0453      10        1    [1,10.9]
454  A0454      52        1 (50.5,60.4]
455  A0455      90        0 (80.2,90.1]
456  A0456      10        1    [1,10.9]
457  A0457      66        0 (60.4,70.3]
458  A0458      89        0 (80.2,90.1]
459  A0459      13        1 (10.9,20.8]
460  A0460      77        0 (70.3,80.2]
461  A0461      84        0 (80.2,90.1]
462  A0462      26        1 (20.8,30.7]
463  A0463      37        1 (30.7,40.6]
464  A0464      17        1 (10.9,20.8]
465  A0465      49        1 (40.6,50.5]
466  A0466       5        1    [1,10.9]
467  A0467      14        1 (10.9,20.8]
468  A0468      43        1 (40.6,50.5]
469  A0469      98        0  (90.1,100]
470  A0470      23        1 (20.8,30.7]
471  A0471      62        0 (60.4,70.3]
472  A0472      10        1    [1,10.9]
473  A0473       3        1    [1,10.9]
474  A0474      99        0  (90.1,100]
475  A0475      21        1 (20.8,30.7]
476  A0476       8        1    [1,10.9]
477  A0477      88        0 (80.2,90.1]
478  A0478      78        0 (70.3,80.2]
479  A0479      56        1 (50.5,60.4]
480  A0480      12        1 (10.9,20.8]
481  A0481      85        0 (80.2,90.1]
482  A0482      18        1 (10.9,20.8]
483  A0483      87        0 (80.2,90.1]
484  A0484      77        0 (70.3,80.2]
485  A0485       1        1    [1,10.9]
486  A0486      60        0 (50.5,60.4]
487  A0487       6        1    [1,10.9]
488  A0488      85        0 (80.2,90.1]
489  A0489      32        0 (30.7,40.6]
490  A0490      88        0 (80.2,90.1]
491  A0491      47        1 (40.6,50.5]
492  A0492      58        0 (50.5,60.4]
493  A0493      36        1 (30.7,40.6]
494  A0494      94        0  (90.1,100]
495  A0495      17        1 (10.9,20.8]
496  A0496      69        1 (60.4,70.3]
497  A0497      60        0 (50.5,60.4]
498  A0498      86        0 (80.2,90.1]
499  A0499       9        0    [1,10.9]
500  A0500      27        1 (20.8,30.7]
501  A0501      58        0 (50.5,60.4]
502  A0502      32        0 (30.7,40.6]
503  A0503       1        1    [1,10.9]
504  A0504      14        1 (10.9,20.8]
505  A0505      93        0  (90.1,100]
506  A0506       6        1    [1,10.9]
507  A0507      43        0 (40.6,50.5]
508  A0508      47        0 (40.6,50.5]
509  A0509      73        0 (70.3,80.2]
510  A0510      31        1 (30.7,40.6]
511  A0511      17        1 (10.9,20.8]
512  A0512       1        1    [1,10.9]
513  A0513      48        1 (40.6,50.5]
514  A0514       9        1    [1,10.9]
515  A0515      68        0 (60.4,70.3]
516  A0516      91        0  (90.1,100]
517  A0517      61        0 (60.4,70.3]
518  A0518      55        0 (50.5,60.4]
519  A0519      15        1 (10.9,20.8]
520  A0520     100        0  (90.1,100]
521  A0521      91        0  (90.1,100]
522  A0522      34        0 (30.7,40.6]
523  A0523      18        1 (10.9,20.8]
524  A0524      67        0 (60.4,70.3]
525  A0525       2        1    [1,10.9]
526  A0526      60        0 (50.5,60.4]
527  A0527      41        0 (40.6,50.5]
528  A0528      50        1 (40.6,50.5]
529  A0529       4        1    [1,10.9]
530  A0530      98        0  (90.1,100]
531  A0531      85        0 (80.2,90.1]
532  A0532      74        1 (70.3,80.2]
533  A0533      69        0 (60.4,70.3]
534  A0534      33        1 (30.7,40.6]
535  A0535      71        0 (70.3,80.2]
536  A0536      68        0 (60.4,70.3]
537  A0537      54        0 (50.5,60.4]
538  A0538      13        1 (10.9,20.8]
539  A0539      32        1 (30.7,40.6]
540  A0540      26        1 (20.8,30.7]
541  A0541      42        0 (40.6,50.5]
542  A0542      28        1 (20.8,30.7]
543  A0543      14        1 (10.9,20.8]
544  A0544      72        0 (70.3,80.2]
545  A0545      68        0 (60.4,70.3]
546  A0546      48        1 (40.6,50.5]
547  A0547      13        1 (10.9,20.8]
548  A0548       6        1    [1,10.9]
549  A0549      51        1 (50.5,60.4]
550  A0550      79        0 (70.3,80.2]
551  A0551      71        0 (70.3,80.2]
552  A0552      37        1 (30.7,40.6]
553  A0553      34        1 (30.7,40.6]
554  A0554      58        1 (50.5,60.4]
555  A0555      76        0 (70.3,80.2]
556  A0556      72        0 (70.3,80.2]
557  A0557      79        0 (70.3,80.2]
558  A0558      17        1 (10.9,20.8]
559  A0559      98        0  (90.1,100]
560  A0560       2        1    [1,10.9]
561  A0561      88        0 (80.2,90.1]
562  A0562      10        1    [1,10.9]
563  A0563      75        0 (70.3,80.2]
564  A0564      96        0  (90.1,100]
565  A0565      10        1    [1,10.9]
566  A0566     100        0  (90.1,100]
567  A0567      42        0 (40.6,50.5]
568  A0568      48        0 (40.6,50.5]
569  A0569       5        1    [1,10.9]
570  A0570      84        0 (80.2,90.1]
571  A0571      25        1 (20.8,30.7]
572  A0572      51        0 (50.5,60.4]
573  A0573      59        0 (50.5,60.4]
574  A0574      89        0 (80.2,90.1]
575  A0575      24        1 (20.8,30.7]
576  A0576      67        0 (60.4,70.3]
577  A0577      90        0 (80.2,90.1]
578  A0578      28        1 (20.8,30.7]
579  A0579      57        0 (50.5,60.4]
580  A0580      48        1 (40.6,50.5]
581  A0581      80        0 (70.3,80.2]
582  A0582      58        0 (50.5,60.4]
583  A0583      65        0 (60.4,70.3]
584  A0584      14        1 (10.9,20.8]
585  A0585      47        0 (40.6,50.5]
586  A0586      92        0  (90.1,100]
587  A0587      95        0  (90.1,100]
588  A0588      93        0  (90.1,100]
589  A0589      81        0 (80.2,90.1]
590  A0590      37        0 (30.7,40.6]
591  A0591      87        0 (80.2,90.1]
592  A0592      56        0 (50.5,60.4]
593  A0593      87        0 (80.2,90.1]
594  A0594      21        1 (20.8,30.7]
595  A0595      35        1 (30.7,40.6]
596  A0596      62        0 (60.4,70.3]
597  A0597      14        1 (10.9,20.8]
598  A0598      62        1 (60.4,70.3]
599  A0599       4        1    [1,10.9]
600  A0600      74        0 (70.3,80.2]
601  A0601      85        0 (80.2,90.1]
602  A0602      94        0  (90.1,100]
603  A0603      39        1 (30.7,40.6]
604  A0604      47        1 (40.6,50.5]
605  A0605      15        1 (10.9,20.8]
606  A0606      27        1 (20.8,30.7]
607  A0607      38        1 (30.7,40.6]
608  A0608      25        1 (20.8,30.7]
609  A0609      71        0 (70.3,80.2]
610  A0610       8        1    [1,10.9]
611  A0611      69        0 (60.4,70.3]
612  A0612      51        0 (50.5,60.4]
613  A0613      81        0 (80.2,90.1]
614  A0614      15        1 (10.9,20.8]
615  A0615       5        1    [1,10.9]
616  A0616      53        0 (50.5,60.4]
617  A0617      60        0 (50.5,60.4]
618  A0618      10        1    [1,10.9]
619  A0619      25        1 (20.8,30.7]
620  A0620      59        1 (50.5,60.4]
621  A0621      82        0 (80.2,90.1]
622  A0622      85        0 (80.2,90.1]
623  A0623      91        0  (90.1,100]
624  A0624      68        0 (60.4,70.3]
625  A0625      83        0 (80.2,90.1]
626  A0626      52        0 (50.5,60.4]
627  A0627      60        0 (50.5,60.4]
628  A0628      26        1 (20.8,30.7]
629  A0629      15        1 (10.9,20.8]
630  A0630      67        0 (60.4,70.3]
631  A0631      91        0  (90.1,100]
632  A0632      25        1 (20.8,30.7]
633  A0633      45        1 (40.6,50.5]
634  A0634      95        0  (90.1,100]
635  A0635      75        1 (70.3,80.2]
636  A0636      48        1 (40.6,50.5]
637  A0637      72        0 (70.3,80.2]
638  A0638      31        1 (30.7,40.6]
639  A0639      88        0 (80.2,90.1]
640  A0640      29        1 (20.8,30.7]
641  A0641      90        0 (80.2,90.1]
642  A0642      26        1 (20.8,30.7]
643  A0643      70        1 (60.4,70.3]
644  A0644      21        1 (20.8,30.7]
645  A0645      86        0 (80.2,90.1]
646  A0646      91        0  (90.1,100]
647  A0647      55        1 (50.5,60.4]
648  A0648      60        0 (50.5,60.4]
649  A0649      36        0 (30.7,40.6]
650  A0650      32        1 (30.7,40.6]
651  A0651      98        0  (90.1,100]
652  A0652      94        0  (90.1,100]
653  A0653      63        0 (60.4,70.3]
654  A0654      63        0 (60.4,70.3]
655  A0655       9        1    [1,10.9]
656  A0656      61        0 (60.4,70.3]
657  A0657      12        1 (10.9,20.8]
658  A0658      40        1 (30.7,40.6]
659  A0659      85        0 (80.2,90.1]
660  A0660      68        1 (60.4,70.3]
661  A0661      98        0  (90.1,100]
662  A0662      21        1 (20.8,30.7]
663  A0663      64        0 (60.4,70.3]
664  A0664      86        0 (80.2,90.1]
665  A0665      77        0 (70.3,80.2]
666  A0666       5        1    [1,10.9]
667  A0667      75        0 (70.3,80.2]
668  A0668      30        0 (20.8,30.7]
669  A0669      96        0  (90.1,100]
670  A0670      94        0  (90.1,100]
671  A0671      50        0 (40.6,50.5]
672  A0672      16        1 (10.9,20.8]
673  A0673      67        0 (60.4,70.3]
674  A0674      77        0 (70.3,80.2]
675  A0675      84        0 (80.2,90.1]
676  A0676       4        1    [1,10.9]
677  A0677      70        0 (60.4,70.3]
678  A0678      74        0 (70.3,80.2]
679  A0679      80        0 (70.3,80.2]
680  A0680      93        0  (90.1,100]
681  A0681      65        0 (60.4,70.3]
682  A0682      28        1 (20.8,30.7]
683  A0683      81        0 (80.2,90.1]
684  A0684      67        0 (60.4,70.3]
685  A0685      60        0 (50.5,60.4]
686  A0686       4        1    [1,10.9]
687  A0687      20        1 (10.9,20.8]
688  A0688      21        1 (20.8,30.7]
689  A0689      73        0 (70.3,80.2]
690  A0690      99        0  (90.1,100]
691  A0691      93        0  (90.1,100]
692  A0692      27        1 (20.8,30.7]
693  A0693      36        1 (30.7,40.6]
694  A0694      36        1 (30.7,40.6]
695  A0695      84        0 (80.2,90.1]
696  A0696      34        1 (30.7,40.6]
697  A0697      55        0 (50.5,60.4]
698  A0698      54        0 (50.5,60.4]
699  A0699      98        0  (90.1,100]
700  A0700      56        0 (50.5,60.4]
701  A0701      21        1 (20.8,30.7]
702  A0702      60        0 (50.5,60.4]
703  A0703      68        0 (60.4,70.3]
704  A0704      29        1 (20.8,30.7]
705  A0705      69        0 (60.4,70.3]
706  A0706      28        1 (20.8,30.7]
707  A0707      99        0  (90.1,100]
708  A0708      24        0 (20.8,30.7]
709  A0709      47        1 (40.6,50.5]
710  A0710       1        1    [1,10.9]
711  A0711      93        0  (90.1,100]
712  A0712      97        0  (90.1,100]
713  A0713       3        1    [1,10.9]
714  A0714      51        1 (50.5,60.4]
715  A0715      91        0  (90.1,100]
716  A0716      27        1 (20.8,30.7]
717  A0717      10        1    [1,10.9]
718  A0718      58        1 (50.5,60.4]
719  A0719      99        0  (90.1,100]
720  A0720      12        1 (10.9,20.8]
721  A0721      13        1 (10.9,20.8]
722  A0722      92        0  (90.1,100]
723  A0723      53        0 (50.5,60.4]
724  A0724       4        1    [1,10.9]
725  A0725      29        1 (20.8,30.7]
726  A0726      24        1 (20.8,30.7]
727  A0727      31        1 (30.7,40.6]
728  A0728      51        0 (50.5,60.4]
729  A0729      89        0 (80.2,90.1]
730  A0730      13        1 (10.9,20.8]
731  A0731      35        0 (30.7,40.6]
732  A0732      72        0 (70.3,80.2]
733  A0733      23        1 (20.8,30.7]
734  A0734      58        0 (50.5,60.4]
735  A0735       9        1    [1,10.9]
736  A0736      67        0 (60.4,70.3]
737  A0737      26        1 (20.8,30.7]
738  A0738      40        1 (30.7,40.6]
739  A0739       6        1    [1,10.9]
740  A0740      76        0 (70.3,80.2]
741  A0741       4        1    [1,10.9]
742  A0742      23        1 (20.8,30.7]
743  A0743      63        0 (60.4,70.3]
744  A0744      56        0 (50.5,60.4]
745  A0745      28        0 (20.8,30.7]
746  A0746      37        1 (30.7,40.6]
747  A0747      65        1 (60.4,70.3]
748  A0748      54        1 (50.5,60.4]
749  A0749      77        0 (70.3,80.2]
750  A0750      57        0 (50.5,60.4]
751  A0751      64        0 (60.4,70.3]
752  A0752      12        1 (10.9,20.8]
753  A0753      22        1 (20.8,30.7]
754  A0754      46        0 (40.6,50.5]
755  A0755      76        0 (70.3,80.2]
756  A0756      17        1 (10.9,20.8]
757  A0757      18        1 (10.9,20.8]
758  A0758      27        1 (20.8,30.7]
759  A0759      17        1 (10.9,20.8]
760  A0760      34        1 (30.7,40.6]
761  A0761      66        0 (60.4,70.3]
762  A0762      58        0 (50.5,60.4]
763  A0763      73        0 (70.3,80.2]
764  A0764      62        1 (60.4,70.3]
765  A0765      57        1 (50.5,60.4]
766  A0766      18        1 (10.9,20.8]
767  A0767      13        1 (10.9,20.8]
768  A0768      12        1 (10.9,20.8]
769  A0769      17        1 (10.9,20.8]
770  A0770      80        0 (70.3,80.2]
771  A0771      84        0 (80.2,90.1]
772  A0772      63        0 (60.4,70.3]
773  A0773       9        1    [1,10.9]
774  A0774      49        1 (40.6,50.5]
775  A0775      22        1 (20.8,30.7]
776  A0776      70        0 (60.4,70.3]
777  A0777      65        1 (60.4,70.3]
778  A0778      44        1 (40.6,50.5]
779  A0779      51        1 (50.5,60.4]
780  A0780      70        0 (60.4,70.3]
781  A0781      52        1 (50.5,60.4]
782  A0782      51        0 (50.5,60.4]
783  A0783       1        1    [1,10.9]
784  A0784      52        0 (50.5,60.4]
785  A0785      17        0 (10.9,20.8]
786  A0786       8        1    [1,10.9]
787  A0787      50        0 (40.6,50.5]
788  A0788      10        1    [1,10.9]
789  A0789      95        0  (90.1,100]
790  A0790      23        1 (20.8,30.7]
791  A0791      36        1 (30.7,40.6]
792  A0792      34        1 (30.7,40.6]
793  A0793      20        1 (10.9,20.8]
794  A0794      38        1 (30.7,40.6]
795  A0795      94        0  (90.1,100]
796  A0796      14        1 (10.9,20.8]
797  A0797       1        1    [1,10.9]
798  A0798      81        0 (80.2,90.1]
799  A0799      69        0 (60.4,70.3]
800  A0800      37        1 (30.7,40.6]
801  A0801      61        0 (60.4,70.3]
802  A0802      12        1 (10.9,20.8]
803  A0803       1        1    [1,10.9]
804  A0804      93        0  (90.1,100]
805  A0805     100        0  (90.1,100]
806  A0806      61        0 (60.4,70.3]
807  A0807       4        1    [1,10.9]
808  A0808       3        1    [1,10.9]
809  A0809      91        0  (90.1,100]
810  A0810      89        0 (80.2,90.1]
811  A0811      14        1 (10.9,20.8]
812  A0812      73        0 (70.3,80.2]
813  A0813      89        0 (80.2,90.1]
814  A0814      47        0 (40.6,50.5]
815  A0815      39        1 (30.7,40.6]
816  A0816      34        1 (30.7,40.6]
817  A0817      26        1 (20.8,30.7]
818  A0818      73        0 (70.3,80.2]
819  A0819      55        1 (50.5,60.4]
820  A0820      47        1 (40.6,50.5]
821  A0821      64        0 (60.4,70.3]
822  A0822      43        1 (40.6,50.5]
823  A0823      47        1 (40.6,50.5]
824  A0824      39        1 (30.7,40.6]
825  A0825      66        0 (60.4,70.3]
826  A0826      20        1 (10.9,20.8]
827  A0827      70        0 (60.4,70.3]
828  A0828      41        1 (40.6,50.5]
829  A0829      90        0 (80.2,90.1]
830  A0830      40        1 (30.7,40.6]
831  A0831       8        1    [1,10.9]
832  A0832      47        0 (40.6,50.5]
833  A0833      97        0  (90.1,100]
834  A0834      93        0  (90.1,100]
835  A0835      19        1 (10.9,20.8]
836  A0836      18        1 (10.9,20.8]
837  A0837      29        1 (20.8,30.7]
838  A0838      55        1 (50.5,60.4]
839  A0839      72        0 (70.3,80.2]
840  A0840      62        0 (60.4,70.3]
841  A0841      84        0 (80.2,90.1]
842  A0842      23        1 (20.8,30.7]
843  A0843       6        1    [1,10.9]
844  A0844      56        0 (50.5,60.4]
845  A0845       3        1    [1,10.9]
846  A0846      20        1 (10.9,20.8]
847  A0847      61        0 (60.4,70.3]
848  A0848      43        1 (40.6,50.5]
849  A0849      38        1 (30.7,40.6]
850  A0850      92        0  (90.1,100]
851  A0851      60        0 (50.5,60.4]
852  A0852      76        0 (70.3,80.2]
853  A0853       1        1    [1,10.9]
854  A0854      44        1 (40.6,50.5]
855  A0855      92        0  (90.1,100]
856  A0856      89        0 (80.2,90.1]
857  A0857       5        1    [1,10.9]
858  A0858      29        1 (20.8,30.7]
859  A0859      55        0 (50.5,60.4]
860  A0860      13        1 (10.9,20.8]
861  A0861      24        1 (20.8,30.7]
862  A0862      58        0 (50.5,60.4]
863  A0863      32        1 (30.7,40.6]
864  A0864      98        0  (90.1,100]
865  A0865      12        1 (10.9,20.8]
866  A0866      21        1 (20.8,30.7]
867  A0867      85        0 (80.2,90.1]
868  A0868      81        0 (80.2,90.1]
869  A0869      16        0 (10.9,20.8]
870  A0870      16        1 (10.9,20.8]
871  A0871      55        1 (50.5,60.4]
872  A0872      45        0 (40.6,50.5]
873  A0873       5        1    [1,10.9]
874  A0874      17        1 (10.9,20.8]
875  A0875      91        0  (90.1,100]
876  A0876      45        1 (40.6,50.5]
877  A0877      45        1 (40.6,50.5]
878  A0878      96        0  (90.1,100]
879  A0879      28        0 (20.8,30.7]
880  A0880      36        1 (30.7,40.6]
881  A0881      99        0  (90.1,100]
882  A0882      27        1 (20.8,30.7]
883  A0883      16        1 (10.9,20.8]
884  A0884      84        0 (80.2,90.1]
885  A0885      80        0 (70.3,80.2]
886  A0886      15        1 (10.9,20.8]
887  A0887      56        0 (50.5,60.4]
888  A0888      22        1 (20.8,30.7]
889  A0889      47        1 (40.6,50.5]
890  A0890      41        1 (40.6,50.5]
891  A0891      52        0 (50.5,60.4]
892  A0892      50        0 (40.6,50.5]
893  A0893      75        0 (70.3,80.2]
894  A0894      63        0 (60.4,70.3]
895  A0895       6        0    [1,10.9]
896  A0896      78        0 (70.3,80.2]
897  A0897      13        1 (10.9,20.8]
898  A0898      41        1 (40.6,50.5]
899  A0899      83        0 (80.2,90.1]
900  A0900      18        1 (10.9,20.8]
901  A0901      52        0 (50.5,60.4]
902  A0902      64        0 (60.4,70.3]
903  A0903       3        1    [1,10.9]
904  A0904      41        0 (40.6,50.5]
905  A0905      65        0 (60.4,70.3]
906  A0906      80        0 (70.3,80.2]
907  A0907      82        0 (80.2,90.1]
908  A0908      58        0 (50.5,60.4]
909  A0909      99        0  (90.1,100]
910  A0910      81        0 (80.2,90.1]
911  A0911      87        1 (80.2,90.1]
912  A0912      89        0 (80.2,90.1]
913  A0913      87        0 (80.2,90.1]
914  A0914      19        1 (10.9,20.8]
915  A0915      78        0 (70.3,80.2]
916  A0916      93        0  (90.1,100]
917  A0917      90        0 (80.2,90.1]
918  A0918      80        0 (70.3,80.2]
919  A0919      38        1 (30.7,40.6]
920  A0920      64        0 (60.4,70.3]
921  A0921       1        1    [1,10.9]
922  A0922      89        0 (80.2,90.1]
923  A0923      92        0  (90.1,100]
924  A0924      35        1 (30.7,40.6]
925  A0925      90        0 (80.2,90.1]
926  A0926      89        0 (80.2,90.1]
927  A0927      53        0 (50.5,60.4]
928  A0928      79        0 (70.3,80.2]
929  A0929      98        0  (90.1,100]
930  A0930      77        0 (70.3,80.2]
931  A0931      97        0  (90.1,100]
932  A0932      97        0  (90.1,100]
933  A0933      87        0 (80.2,90.1]
934  A0934      74        0 (70.3,80.2]
935  A0935      52        1 (50.5,60.4]
936  A0936      33        1 (30.7,40.6]
937  A0937      82        0 (80.2,90.1]
938  A0938      87        0 (80.2,90.1]
939  A0939      79        0 (70.3,80.2]
940  A0940      85        0 (80.2,90.1]
941  A0941      86        0 (80.2,90.1]
942  A0942      73        0 (70.3,80.2]
943  A0943      96        0  (90.1,100]
944  A0944      14        1 (10.9,20.8]
945  A0945       7        1    [1,10.9]
946  A0946      67        0 (60.4,70.3]
947  A0947      89        0 (80.2,90.1]
948  A0948      30        1 (20.8,30.7]
949  A0949      94        0  (90.1,100]
950  A0950      92        0  (90.1,100]
951  A0951      47        1 (40.6,50.5]
952  A0952      23        0 (20.8,30.7]
953  A0953      83        0 (80.2,90.1]
954  A0954      63        0 (60.4,70.3]
955  A0955      69        1 (60.4,70.3]
956  A0956      11        1 (10.9,20.8]
957  A0957      10        1    [1,10.9]
958  A0958      26        1 (20.8,30.7]
959  A0959      32        0 (30.7,40.6]
960  A0960      97        0  (90.1,100]
961  A0961       5        1    [1,10.9]
962  A0962      44        1 (40.6,50.5]
963  A0963     100        0  (90.1,100]
964  A0964      46        1 (40.6,50.5]
965  A0965      92        0  (90.1,100]
966  A0966      23        1 (20.8,30.7]
967  A0967      11        1 (10.9,20.8]
968  A0968      49        1 (40.6,50.5]
969  A0969      29        1 (20.8,30.7]
970  A0970      63        0 (60.4,70.3]
971  A0971      87        0 (80.2,90.1]
972  A0972      42        1 (40.6,50.5]
973  A0973       8        1    [1,10.9]
974  A0974      33        1 (30.7,40.6]
975  A0975      21        1 (20.8,30.7]
976  A0976      78        0 (70.3,80.2]
977  A0977       1        1    [1,10.9]
978  A0978      28        1 (20.8,30.7]
979  A0979      63        0 (60.4,70.3]
980  A0980      11        1 (10.9,20.8]
981  A0981      19        1 (10.9,20.8]
982  A0982      14        0 (10.9,20.8]
983  A0983      76        0 (70.3,80.2]
984  A0984      29        1 (20.8,30.7]
985  A0985       8        1    [1,10.9]
986  A0986      25        1 (20.8,30.7]
987  A0987      90        0 (80.2,90.1]
988  A0988      59        0 (50.5,60.4]
989  A0989      52        0 (50.5,60.4]
990  A0990      19        1 (10.9,20.8]
991  A0991      56        0 (50.5,60.4]
992  A0992      97        0  (90.1,100]
993  A0993       9        1    [1,10.9]
994  A0994      11        1 (10.9,20.8]
995  A0995       9        1    [1,10.9]
996  A0996      98        0  (90.1,100]
997  A0997       6        1    [1,10.9]
998  A0998       9        1    [1,10.9]
999  A0999      17        1 (10.9,20.8]
1000 A1000      15        1 (10.9,20.8]

Learn by Taking It Apart

singers |>
  mutate(bin = cut_interval(quality, 10)) |>
  summary()

       id         quality         SPLATTED              bin     
 A0001  :  1   Min.   :  1.0   Min.   :0.000   (90.1,100] :114  
 A0002  :  1   1st Qu.: 26.0   1st Qu.:0.000   (80.2,90.1]:112  
 A0003  :  1   Median : 55.0   Median :0.000   [1,10.9]   :109  
 A0004  :  1   Mean   : 52.1   Mean   :0.452   (60.4,70.3]:104  
 A0005  :  1   3rd Qu.: 78.0   3rd Qu.:1.000   (50.5,60.4]:103  
 A0006  :  1   Max.   :100.0   Max.   :1.000   (70.3,80.2]:103  
 (Other):994                                   (Other)    :355

Learn by Taking It Apart

singers |>
  mutate(bin = cut_interval(quality, 10)) |>
  group_by(bin) |>
  summarise(prop = mean(SPLATTED))

# A tibble: 10 × 2
   bin            prop
   <fct>         <dbl>
 1 [1,10.9]    0.982  
 2 (10.9,20.8] 0.959  
 3 (20.8,30.7] 0.935  
 4 (30.7,40.6] 0.803  
 5 (40.6,50.5] 0.573  
 6 (50.5,60.4] 0.311  
 7 (60.4,70.3] 0.115  
 8 (70.3,80.2] 0.0388 
 9 (80.2,90.1] 0.00893
10 (90.1,100]  0.0351

Never Trust Yourself

singers <- read_csv('some file.csv')

check and continue

if (nrow(singers != 1000)) {
  warning("unexpected number of rows -> ", nrow(singers))
}

Warning: unexpected number of rows -> 999

check and stop

if (nrow(singers != 1000)) {
  stop("unexpected number of rows -> ", nrow(singers))
}

Don’t Trust Magic

{sjPlot}, {effects} etc.

set some parameters
instant pretty output

{broom}, {ggplot2}, etc.

intermediate calculations
work to make pretty output

more likely to understand where the numbers come from

Want to Play?

reanalyse your own undergraduate data

library(help="haven") # tools for reading SPSS data

play with other datasets
many datasets built in to R and its libraries

data()

Want to Learn?

Oscar Baruffa’s Big Book of R https://bigbookofr.com

Towards Stats Ninja

worth thinking about how far you’ve come
- got to grips with stats, graphics
  \(t\), \(F\), \(\chi^2\), \(R^2\), intercepts, slopes, logits, confidence intervals…
- got to grips with programming variables, functions, {tidyverse}, {ggplot2}, libraries, RStudio…
- playing with .Rmd etc

please feed back (good or bad)

A Take-Home Message: ‘Many Analysts’

Twenty-nine teams involving 61 analysts used the same data set to address the same research question: whether soccer referees are more likely to give red cards to dark-skin-toned players than to light-skin-toned players. Analytic approaches varied widely across the teams, and the estimated effect sizes ranged from 0.89 to 2.93 (Mdn = 1.31) in odds-ratio units. Twenty teams (69%) found a statistically significant positive effect, and 9 teams (31%) did not observe a significant relationship.

Silberzahn, R., et al. (2018). https://doi.org/10/gd2429

statistics is a language for talking about patterns in data

Some Kind of End to the Course

Zen and the Art of Stats

Null-Hypothesis Statistical Testing

Null-Hypothesis Statistical Testing

Null-Hypothesis Statistical Testing

A Theory about the Structure of Chance

What is Enough?

But Linear Models…?

Coefficients

when predictors are categories

Non-Linear Outcomes

Non-Linear Outcomes

Model Building

When Should I Put Things in My Models?

Control Isn’t Always Easy

Control Isn’t Always Easy

Control Isn’t Always Easy

Conclusions Aren’t Always Easy

Conclusions Aren’t Always Easy

Conclusions Aren’t Always Easy

The Bits We Didn’t Talk About

ANOVA

History

Multiple Regression

ANOVA

Why Teach GLM/Regression?

ANOVA in R

GLM vs ANOVA

GLM vs ANOVA

The Elephant in the Room

Repeated Measures

Mixed-Effects Models

Reaching Nirv-R-na

R is a Language

Learn by Taking It Apart

Learn by Taking It Apart

Learn by Taking It Apart

Learn by Taking It Apart

Never Trust Yourself

check and continue

check and stop

Don’t Trust Magic

Want to Play?

Want to Learn?

Towards Stats Ninja

A Take-Home Message: ‘Many Analysts’

Goodbye!