The SPINE of statistics

# The SPINE of statistics: Standard error and confidence intervals
## Professor Andy Field

<div>
  <img style="vertical-align:middle; width:30px; height:30px" src="media/twitter_60.png">
  <span style="line-height:40px;">@profandyfield</span>
</div>
<div>
  <img style="vertical-align:middle; width:60px" src="media/youtube.png">
  <span style="line-height:40px;">www.youtube.com/user/ProfAndyField/</span>
</div>
<div>
  <img style="vertical-align:middle; width:30px; height:30px" src="media/ds_com_fav.png">
  <span style="line-height:40px;">www.discoveringstatistics.com</span>
</div>
<div>
  <img style="vertical-align:middle; width:30px; height:30px" src="media/milton_grey_fav.png">
  <span style="line-height:40px;">www.milton-the-cat.rocks</span>
</div>
<div>
  <img style="vertical-align:middle; width:30px; height:30px" src="media/discovr_fav.png">
  <span style="line-height:40px;">www.discovr.rocks</span>
</div>

???
    h or ?: Toggle the help window
    j: Jump to next slide
    k: Jump to previous slide
    b: Toggle blackout mode
    m: Toggle mirrored mode.
    p: Toggle PresenterMode
    f: Toggle Fullscreen
    t: Reset presentation timer
    <number> + <Return>: Jump to slide <number>
    c: Create a clone presentation on a new window

---
# The SPINE of statistics

## 5 Key concepts

* **S**tandard error

* **P**arameters

* **I**nterval estimates
 
* **N**ull hypothesis significance testing (NHST)

* **E**stimation

---

![:scale 83%](media/spine_content_map.png)
???

We've seen this map of the process of fitting models before

---
class: center

![:scale 83%](media/spine_lec_02.png)

???

Today we focus on why we sample, and what it means, and look in detail at the concepts of the standard error and confidence interval.

---
#  Learning outcomes

* Understand what sampling error is

* Understand what the standard error represents

* Understand what a confidence interval represents
  + and also what it does NOT represent!
  
--

* Understand that parameter estimates are raw effect sizes
  
--

* Be able to interpret
  + Parameter estimates
  + Their confidence intervals
  + Standardized parameter estimates

---
# The only equation you will ever need
## The General Linear Model (GLM)
--

.ong_dk[
$$
`\begin{aligned}
\text{outcome}_i &= (\text{model}_i) + \text{error}_i \\
\text{outcome}_i &= \hat{b}_0 + \hat{b}_1\text{predictor}_{i} + \dots + \hat{b}_n\text{predictor}_{i} + \text{error}_i
\end{aligned}`
$$
]

`$\hat{b}_n$`

* Estimate of parameter for a predictor
	+ Direction/strength of relationship/effect
	+ Difference in means

`$\hat{b}_0$`

* Estimate of the value of the outcome when predictor(s) = 0 (intercept)

---
.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{volume}_{i} + e_i
\end{aligned}`
$$
]

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{volume}_{i} + e_i
\end{aligned}`
$$
]

--
.pull-left[
![](spine_of_statistics_files/figure-html/unnamed-chunk-11-1.png)
]

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{volume}_{i} + e_i
\end{aligned}`
$$
]

???

Let's zoom out of the plot a bit
---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{volume}_{i} + e_i
\end{aligned}`
$$
]

---
class: center

.ong_dk[
$$
`\begin{aligned}
\hat{\text{ringing}}_i &= -37.12 + 10.45\text{volume}_{i}
\end{aligned}`
$$
]

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{musician}_{i} + e_i
\end{aligned}`
$$
]

![](spine_of_statistics_files/figure-html/unnamed-chunk-17-1.png)

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{musician}_{i} + e_i
\end{aligned}`
$$
]

![](spine_of_statistics_files/figure-html/unnamed-chunk-18-1.png)

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{musician}_{i} + e_i
\end{aligned}`
$$
]

![](spine_of_statistics_files/figure-html/unnamed-chunk-19-1.png)

---
class: center, middle

![:scale 100%](media/dsr2_fig_06_04_lm_sample_01.png)
---
class: center, middle

![:scale 100%](media/dsr2_fig_06_04_lm_sample_02.png)

---
class: center, middle

![:scale 100%](media/dsr2_fig_06_04_lm_sample_03.png)

---
class: center

# A test-y example

![:scale 50%](media/andy_zoe_as_fisher_nightingale_2015.jpg)
--

![:abs 25%, 75%, 10%](media/ronald_fisher.png)
![:abs 25%, 0%, 10%](media/florence_nightingale.png)
---

# Sampling error

![:abs 40%, 30%, 5%](media/pop.png)

![:abs 15%, 10%, 40%](media/sample_1.png)
![:abs 9%, 23%, 27%](media/arrow_1.png)

![:abs 15%, 25%, 52%](media/sample_2.png)
![:abs 4%, 33%, 40%](media/arrow_2.png)

![:abs 15%, 40%, 60%](media/sample_3.png)
![:abs 3%, 47%, 48%](media/arrow_3.png)

![:abs 15%, 55%, 52%](media/sample_4.png)
![:abs 7%, 55%, 45%](media/arrow_5.png)

![:abs 15%, 70%, 40%](media/sample_5.png)
![:abs 10%, 67%, 30%](media/arrow_5.png)

---
background-image: none

# Standard error

![:abs 40%, 40%, 5%](media/ais_standard_error.png)

---
background-image: none

![:abs 75%, 12%, 5%](media/ais_c08_standard_error_comparisons_sketch.png)
---

# The Central Limit Theorem (CLT)

---
# The Central Limit Theorem (CLT)

---
# The Central Limit Theorem (CLT)

---
background-image: none

# Confidence interval

![:abs 40%, 5%, 15%](media/ci_pop.png)

![:abs 15%, 37%, 25%](media/ci_arrow.png)

---
background-image: none

# Confidence interval

![:abs 40%, 5%, 15%](media/ci_pop.png)

![:abs 15%, 37%, 25%](media/ci_arrow.png)

---

# Confidence interval

![:abs 40%, 5%, 15%](media/ci_pop.png)

![:abs 15%, 37%, 25%](media/ci_arrow.png)

---

# Confidence interval

![:abs 40%, 5%, 15%](media/ci_pop.png)

![:abs 15%, 37%, 25%](media/ci_arrow.png)

![:abs 60%, 20%, 80%](media/ci_eq.png)

---
background-image: none
background-color: #000000
class: no-scribble

---
# Confidence intervals

.tip[
## <svg style="height:1.5em; top:.04em; position: relative; fill: #2C5577;" viewBox="0 0 640 512"><path d="M512,176a16,16,0,1,0-16-16A15.9908,15.9908,0,0,0,512,176ZM576,32.72461V32l-.46094.3457C548.81445,12.30469,515.97461,0,480,0s-68.81445,12.30469-95.53906,32.3457L384,32v.72461C345.35156,61.93164,320,107.82422,320,160c0,.38086.10938.73242.11133,1.11328A272.01015,272.01015,0,0,0,96,304.26562V176A80.08413,80.08413,0,0,0,16,96a16,16,0,0,0,0,32,48.05249,48.05249,0,0,1,48,48V432a80.08413,80.08413,0,0,0,80,80H352a32.03165,32.03165,0,0,0,32-32,64.0956,64.0956,0,0,0-57.375-63.65625L416,376.625V480a32.03165,32.03165,0,0,0,32,32h32a32.03165,32.03165,0,0,0,32-32V316.77539A160.036,160.036,0,0,0,640,160C640,107.82422,614.64844,61.93164,576,32.72461ZM480,32a126.94015,126.94015,0,0,1,68.78906,20.4082L512,80H448L411.21094,52.4082A126.94015,126.94015,0,0,1,480,32Zm64,64v64a64,64,0,0,1-128,0V96l21.334,16h85.332ZM480,480H448V351.99609A15.99929,15.99929,0,0,0,425.5,337.377L303.1875,391.75a100.1169,100.1169,0,0,0-67.25-84.89062,7.96929,7.96929,0,0,0-10.09375,5.76562l-3.875,15.5625a8.16346,8.16346,0,0,0,5.375,9.5625C252,346.875,272,375.625,272,401.90625V448h48a32.03165,32.03165,0,0,1,32,32H144c-26.94531,0-48.13086-22.27344-47.99609-49.21875.63671-127.52734,101.31054-231.53516,227.36914-238.14063A160.02931,160.02931,0,0,0,480,320Zm0-192A128.14414,128.14414,0,0,1,352,160c0-32.16992,12.334-61.25391,32-83.76367V160a96,96,0,0,0,192,0V76.23633C595.666,98.74609,608,127.83008,608,160A128.14414,128.14414,0,0,1,480,288ZM432,160a16,16,0,1,0,16-16A15.9908,15.9908,0,0,0,432,160ZM162.94531,68.76953l39.71094,16.56055,16.5625,39.71094a5.32345,5.32345,0,0,0,9.53906,0l16.5586-39.71094,39.71484-16.56055a5.336,5.336,0,0,0,0-9.541l-39.71484-16.5586L228.75781,2.957a5.325,5.325,0,0,0-9.53906,0l-16.5625,39.71289-39.71094,16.5586a5.336,5.336,0,0,0,0,9.541Z"/></svg> What they are:
* Intervals that contain the ‘true’ population value of the parameter in 95% of samples.
]
<br>
--

.warning[
## <svg style="height: 1em; top:.04em; position: relative; fill: #CA3E34;" viewBox="0 0 576 512"><path d="M192,320h32V224H192Zm160,0h32V224H352ZM544,112H512a32.03165,32.03165,0,0,0-32,32v16H416V128h32a32.03165,32.03165,0,0,0,32-32V64a32.03165,32.03165,0,0,0-32-32H416a32.03165,32.03165,0,0,0-32,32H352a32.03165,32.03165,0,0,0-32,32v32H256V96a32.03165,32.03165,0,0,0-32-32H192a32.03165,32.03165,0,0,0-32-32H128A32.03165,32.03165,0,0,0,96,64V96a32.03165,32.03165,0,0,0,32,32h32v32H96V144a32.03165,32.03165,0,0,0-32-32H32A32.03165,32.03165,0,0,0,0,144V288a32.03165,32.03165,0,0,0,32,32H64v32a32.03165,32.03165,0,0,0,32,32h32v64a32.03165,32.03165,0,0,0,32,32h80a32.03165,32.03165,0,0,0,32-32V416a32.03165,32.03165,0,0,0-32-32h96a32.03165,32.03165,0,0,0-32,32v32a32.03165,32.03165,0,0,0,32,32h80a32.03165,32.03165,0,0,0,32-32V384h32a32.03165,32.03165,0,0,0,32-32V320h32a32.03165,32.03165,0,0,0,32-32V144A32.03165,32.03165,0,0,0,544,112ZM416,64h32V96H416ZM128,96V64h32V96ZM240,448H160V384h32v32h48Zm176,0H336V416h48V384h32ZM544,288H480v64H96V288H32V144H64V256H96V192h96V96h32v64H352V96h32v96h96v64h32V144h32Z"/></svg> .red[What they are not:]
* There is not a 95% probability that a given interval contains the population value.
    - It is *p* = 0 or *p* = 1, but you can’t know which!
* They do not reflect confidence in the value of the population parameter.
]

---

# Interpreting parameter estimates
## Raw effect size (*b*)

.right-column[
* Does playing hard to get work?<sup>1</sup>
* Heterosexual participants conversed with an opposite-sex confederate over Instant
Messenger for 8 mins
* **Interest**: Final message coded for the number of expressions of romantic interest (range 0 to 4)
* **Hard to get**: 3 items rated 1 (not at all) and 5 (very much so)
  - *The other participant is hard to get*
* **Mate value**: 4 items rated 1 (not at all) and 5 (very much so)
  - *I perceive the other participant as a valued mate*
]

.footnote[
[1] [Birnbaum et al. (2020). *Journal of Social and Personal Relationships*. Study 3.]( https://doi.org/10.1177/0265407520927469)
]
---

.center[
.ong_dk[
.eq_lrge[
`$\text{interest}_i = \hat{b}_0 + \hat{b}_1\text{hard to get}_i +e_i$`
]
]
]

]

.pull-right[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;"> (Intercept) </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 0.310 </td>
   <td style="text-align:right;background-color: white !important;"> 0.524 </td>
   <td style="text-align:right;background-color: white !important;"> 0.591 </td>
   <td style="text-align:right;background-color: white !important;"> 0.556 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> hard_to_get </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.294 </td>
   <td style="text-align:right;"> 0.167 </td>
   <td style="text-align:right;"> 1.765 </td>
   <td style="text-align:right;"> 0.080 </td>
  </tr>
</tbody>
</table>
]

--
* As the perception that the other person was hard to get increased by 1 (on a scale from 1-5), **0.294** more expressions of interest were made.

--
* You'd need perceptions of 'hard to get' to increase by `$\frac{1}{0.294} = 3.4$` on a 5-point scale to get 1 additional expression of interest.

---

# Parameter estimates: 95% confidence interval

.center[
.ong_dk[
.eq_lrge[
`$\text{interest}_i = \hat{b}_0 + \hat{b}_1\text{hard to get}_i +e_i$`
]
]

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
   <th style="text-align:right;"> conf.low </th>
   <th style="text-align:right;"> conf.high </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;"> (Intercept) </td>
   <td style="text-align:right;background-color: white !important;"> 0.310 </td>
   <td style="text-align:right;background-color: white !important;"> 0.524 </td>
   <td style="text-align:right;background-color: white !important;"> 0.591 </td>
   <td style="text-align:right;background-color: white !important;"> 0.556 </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> -0.728 </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 1.347 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> hard_to_get </td>
   <td style="text-align:right;"> 0.294 </td>
   <td style="text-align:right;"> 0.167 </td>
   <td style="text-align:right;"> 1.765 </td>
   <td style="text-align:right;"> 0.080 </td>
   <td style="text-align:right;background-color: yellow !important;"> -0.036 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.624 </td>
  </tr>
</tbody>
</table>

]

* **Assuming that this sample is one of the 95% that yields a confidence interval containing the true value of the parameter ...**

--
* As the perception that the other person was hard to get increases by 1, the corresponding change in the number of expressions made could be as small as **-0.036**. In other words, there are *fewer* expressions of interest.

--
* As the perception that the other person was hard to get increases by 1, the corresponding change in the number of expressions made could be as large as **0.624**. In other words, there are *more* expressions of interest.

--
* It's plausible that as the perception that the other person was hard to get increases by 1, there is *no change* in expressions of interest (*b* = 0).

--
* **The assumption at the start might be false**.
---
background-image: none
background-color: #000000
class: no-scribble

---

# Interpreting parameter estimates
## Standardized effect size `$\beta$`

.pull-left[
* Who believes fake news?<sup>1</sup>
* **fake_newz**: Belief in fake news
  - Average rating of 12 fake news items (1 = Not at all accurate, 4 = Very accurate)
* **delusionz**: Peter's delusion inventory
  - *Do you ever feel as if thereis a conspiracy against you?*
* **crit_thinkz**: Critical thinking
  - 7 problems that have intuitive-but-incorrect responses that must be overridden to arrive at the correct answer
  - *How many cubic feet of dirt are there in a hole that is 3 feet deep by 3 feet wide by 3 feet long?* 
]

.footnote[
[1] [Bronstein et al. (2019). *Journal of Applied Research in Memory and Cognition*.](https://doi.org/10.1016/j.jarmac.2018.09.005)
]

???
CRT task - intuitive answer is 27 cubic feet (3 * 3 * 3), but there are 0 cubic feet of dirt in a hole.

---

.center[
.ong_dk[
.eq_lrge[
`$\text{fake news beliefs}_i = \hat{\beta}_0 + \hat{\beta}_1\text{delusion}_i +e_i$`
]
]
]

]

.pull-right[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;"> (Intercept) </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 0.000 </td>
   <td style="text-align:right;background-color: white !important;"> 0.032 </td>
   <td style="text-align:right;background-color: white !important;"> 0.000 </td>
   <td style="text-align:right;background-color: white !important;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> delusionz </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.242 </td>
   <td style="text-align:right;"> 0.032 </td>
   <td style="text-align:right;"> 7.676 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>
]

--
* For every standard deviation change in delusion proneness, belief in fake news increases by **0.242** standard deviations.
---
# Summary

## You can understand most psychological statistics with 5 concepts

1. Parameters define the model and represent hypotheses of interest
2. Estimation (parameters are estimated based on sample data)
3. NHST (next lecture)
4. Interval estimates
    - All parameters have one
    - Confidence intervals might tell us something about the true value of the parameter
5. Standard Error (of a parameter)
    - Tells us about the variability in parameter estimates from sample to sample
    - Significance tests and confidence intervals rely on the standard error