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"<div class=mystyle1>statistical-modelling::1-introduction</div>
<br>
What is the simple linear regression model?&nbsp;""\(Y_i = β_0 + β_1x_i + ε_i
, \text{ where } i = 1, 2, . . . , n\)&nbsp;<br>where&nbsp;\(\varepsilon _i\)&nbsp;is a random error (or departures from the mean)<br><img src=""Screenshot 2024-03-15 at 8.00.36 pm.png"">"
"<div class=mystyle1>statistical-modelling::1-introduction</div>
<br>
What is the normal simple linear regression model?&nbsp;""<img src=""Screenshot 2024-03-15 at 8.01.22 pm.png"">"
"<div class=mystyle1>statistical-modelling::1-introduction</div>
<br>
What is the multiple linear regression model?&nbsp;""<img src=""Screenshot 2024-03-15 at 8.05.15 pm.png"">"
"<div class=mystyle1>statistical-modelling::1-introduction</div>
<br>
What is the formula for Var(<b>z</b>)""<img src=""Screenshot 2024-03-15 at 8.08.11 pm.png""><br><img src=""Screenshot 2024-03-15 at 8.09.09 pm.png"">"
"<div class=mystyle1>statistical-modelling::estimation</div>
<br>
What is the least squares estimator if there are two model parameters&nbsp;\(β_0\)&nbsp;and&nbsp;\(β_1\)?""minimize the sum of squares
of errors denoted by&nbsp;\(S(β_0, β_1)\). That is, the estimators minimize<br><img src=""Screenshot 2024-03-15 at 8.16.10 pm.png"">"
"<div class=mystyle1>statistical-modelling::estimation</div>
<br>
What is the formula of \(\hat \beta_0\)&nbsp;which gives the smallest&nbsp;\(S(β_0, β_1)\)?"\(\hat \beta_0 = \bar Y− \hat β_1 \bar x\)
"<div class=mystyle1>statistical-modelling::estimation</div>
<br>
What is the formula of&nbsp;\(\hat \beta_1\)&nbsp;which gives the smallest&nbsp;\(S(β_0, β_1)\)?""<img src=""Screenshot 2024-03-15 at 8.24.57 pm.png""><br>is also&nbsp;\(\hat \beta_1 = \frac{(\sum x_iy_i)-n\bar x \bar y}{\sum x_i^2-n \bar x^2}\)&nbsp;"
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"<div class=mystyle1>statistical-modelling::estimation</div>
<br>
What is the formula for the matrix form of the least square estimator&nbsp;\(\mathbf {\hat \beta}\)?&nbsp;""<img src=""Screenshot 2024-03-15 at 8.31.57 pm.png""><br>since<br><img src=""Screenshot 2024-03-15 at 8.32.00 pm.png"">"
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"<div class=mystyle1>statistical-modelling::estimation</div>
<br>
What is the&nbsp;Best Linear Unbiased Estimator (BLUE)?""It is the estimator&nbsp;\(l^T\hat β\), that among all unbiased linear estimators of the form <b>c^TY</b> , has the
smallest variance"
"<div class=mystyle1>statistical-modelling::inference</div>
<br>
What are residuals? (e)""\(e_i
= y_i - x_i \hat\beta = y_i − \hat y_i
, i = 1, . . . , n,\)<br>estimator for the random errors εi&nbsp;<br><br><img src=""Screenshot 2024-03-17 at 1.29.14 pm.png""><br><br><img src=""Screenshot 2024-03-17 at 1.33.29 pm.png""><br>"
"<div class=mystyle1>statistical-modelling::inference</div>
<br>
<div><div>What is the&nbsp;Residual Sum of Squares (SS_E)?&nbsp;</div></div>""<img src=""Screenshot 2024-03-17 at 12.21.42 pm.png""><br><img src=""Screenshot 2024-03-17 at 12.29.56 pm.png""><br><img src=""Screenshot 2024-03-17 at 1.29.53 pm.png"">"
"<div class=mystyle1>statistical-modelling::inference</div>
<br>
What is the&nbsp;Total Sum of Squares SS_T?&nbsp;""<img src=""Screenshot 2024-03-17 at 12.18.25 pm.png""><br><img src=""Screenshot 2024-03-17 at 12.21.42 pm.png""><br><img src=""Screenshot 2024-03-17 at 2.10.51 pm.png"">"
"<div class=mystyle1>statistical-modelling::inference</div>
<br>
What is the&nbsp;Total Sum of Squares SS_T for a linear model?&nbsp;"SS_T = SS_R + SS_E&nbsp;<br>total variation = explained variation + residual variation
"<div class=mystyle1>statistical-modelling::inference</div>
<br>
What is the&nbsp;Total Sum of Squares SS_T&nbsp;for a constant model?&nbsp;"SS_T = SS_E&nbsp;
"<div class=mystyle1>statistical-modelling::inference</div>
<br>
What is the regression sum of squares? SS_R""<img src=""Screenshot 2024-03-17 at 12.21.13 pm.png""><br><img src=""Screenshot 2024-03-17 at 12.21.42 pm.png""><br><img src=""Screenshot 2024-03-17 at 2.10.57 pm.png"">"
"<div class=mystyle1>statistical-modelling::estimation</div>
<br>
What is the hat matrix H?&nbsp;"\(H = X(X^TX)^{-1}X^T\)
"<div class=mystyle1>statistical-modelling::estimation</div>
<br>
What are some properties of I-H? (2)"- it is symmetric<br>- it is idempotent
"<div class=mystyle1>statistical-modelling::estimation</div>
<br>
What is E(SS_E)?&nbsp;"\((n-p) \sigma ^2\)&nbsp;where p is the number of \beta terms
"
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if (mask === null || mask.complete) {
loaded();
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mask.addEventListener('load', loaded);
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</script>
""
<div id=""io-header"">Anova Table</div>
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original.style.visibility = ""visible"";
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if (mask === null || mask.complete) {
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</script>
"
"
<div id=""io-header"">Anova Table</div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""6d156bdc11884058b466d6ac8f4424b4-ao-4-Q.svg"" /></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 12.33.31 pm.png"" /></div>
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if (mask === null || mask.complete) {
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</script>
""
<div id=""io-header"">Anova Table</div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""6d156bdc11884058b466d6ac8f4424b4-ao-4-A.svg""></div>
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</div>
<button id=""io-revl-btn"" onclick=""toggle();"">Toggle Masks</button>
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<script>
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var toggle = function() {
var amask = document.getElementById('io-overlay');
if (amask.style.display === 'block' || amask.style.display === '')
amask.style.display = 'none';
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amask.style.display = 'block'
}
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aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
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var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
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if (mask === null || mask.complete) {
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"
"
<div id=""io-header"">Anova Table</div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""6d156bdc11884058b466d6ac8f4424b4-ao-5-Q.svg"" /></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 12.33.31 pm.png"" /></div>
</div>
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<script>
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original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
""
<div id=""io-header"">Anova Table</div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""6d156bdc11884058b466d6ac8f4424b4-ao-5-A.svg""></div>
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if (amask.style.display === 'block' || amask.style.display === '')
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amask.style.display = 'block'
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aFade = 50, qFade = 0;
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var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
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</script>
"
"
<div id=""io-header"">Anova Table</div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""6d156bdc11884058b466d6ac8f4424b4-ao-6-Q.svg"" /></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 12.33.31 pm.png""></div>
</div>
<div id=""io-footer""></div>
<script>
// Prevent original image from loading before mask
aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
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var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
""
<div id=""io-header"">Anova Table</div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""6d156bdc11884058b466d6ac8f4424b4-ao-6-A.svg""></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 12.33.31 pm.png""></div>
</div>
<button id=""io-revl-btn"" onclick=""toggle();"">Toggle Masks</button>
<div id=""io-extra-wrapper"">
<div id=""io-extra"">
</div>
</div>
<script>
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var toggle = function() {
var amask = document.getElementById('io-overlay');
if (amask.style.display === 'block' || amask.style.display === '')
amask.style.display = 'none';
else
amask.style.display = 'block'
}
// Prevent original image from loading before mask
aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
function loaded() {
var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
"
"<div class=mystyle1>statistical-modelling::estimation</div>
<br>
What is the mean squares (MS) in ANOVA table?&nbsp;"measures of average variation
"<div class=mystyle1>statistical-modelling::estimation</div>
<br>
What is the variance ratio (VR) in ANOVA table?&nbsp;""measures the variation explained by the model fit relative to the variation due to
residuals<br><img src=""Screenshot 2024-03-17 at 12.37.32 pm.png"">"
"<div class=mystyle1>statistical-modelling::1-introduction</div>
<br>
What is the general linear model?&nbsp;""<img src=""Screenshot 2024-03-17 at 1.00.10 pm.png""><br><img src=""Screenshot 2024-03-17 at 1.01.24 pm.png"">"
"<div class=mystyle1>statistical-modelling::inference</div>
<br>
Give an unbiased estimator of the variance using SS_E&nbsp;&nbsp;(residual sum of squares)<br>What is it called?&nbsp;"\(\frac{SS_E}{n-p}\)<br>this is the mean square error/ residual mean square (MSE)
"<div class=mystyle1>statistical-modelling::inference</div>
<br>
What distribution does residuals exhibit?&nbsp;""<img src=""Screenshot 2024-03-17 at 1.33.51 pm.png""><br><img src=""Screenshot 2024-03-19 at 2.12.31 pm.png"">"
"
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""c50213ec02964faab4603d635cf17064-ao-1-Q.svg""></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.35.43 pm.png""></div>
</div>
<div id=""io-footer""></div>
<script>
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aFade = 50, qFade = 0;
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var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
""
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""c50213ec02964faab4603d635cf17064-ao-1-A.svg""></div>
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if (amask.style.display === 'block' || amask.style.display === '')
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aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
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var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
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mask.addEventListener('load', loaded);
}
</script>
"
"
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""c50213ec02964faab4603d635cf17064-ao-2-Q.svg""></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.35.43 pm.png""></div>
</div>
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<script>
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aFade = 50, qFade = 0;
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var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
""
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""c50213ec02964faab4603d635cf17064-ao-2-A.svg""></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.35.43 pm.png""></div>
</div>
<button id=""io-revl-btn"" onclick=""toggle();"">Toggle Masks</button>
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<script>
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var amask = document.getElementById('io-overlay');
if (amask.style.display === 'block' || amask.style.display === '')
amask.style.display = 'none';
else
amask.style.display = 'block'
}
// Prevent original image from loading before mask
aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
function loaded() {
var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
"
"For an idempotent matrix A
<br><span class=""cloze-inactive"" data-ordinal=""2"">trace(A)</span> = <span class=""cloze"" data-cloze=""rank&#x28;A&#x29;"" data-ordinal=""1"">[...]</span><br>
""For an idempotent matrix A
<br><span class=""cloze-inactive"" data-ordinal=""2"">trace(A)</span> = <span class=""cloze"" data-ordinal=""1"">rank(A)</span><br>
"
"For an idempotent matrix A
<br><span class=""cloze"" data-cloze=""trace&#x28;A&#x29;"" data-ordinal=""2"">[...]</span> = <span class=""cloze-inactive"" data-ordinal=""1"">rank(A)</span><br>
""For an idempotent matrix A
<br><span class=""cloze"" data-ordinal=""2"">trace(A)</span> = <span class=""cloze-inactive"" data-ordinal=""1"">rank(A)</span><br>
"
"rank(I − H) = <span class=""cloze"" data-cloze=""n&#x20;−&#x20;p"" data-ordinal=""1"">[...]</span><br>
""rank(I − H) = <span class=""cloze"" data-ordinal=""1"">n − p</span><br>
"
"
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""fe5ba200c55b4e09800cd526480c5d30-oa-1-Q.svg""></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.39.42 pm.png""></div>
</div>
<div id=""io-footer"">if it is a normal linear model</div>
<script>
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aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
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var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
""
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""fe5ba200c55b4e09800cd526480c5d30-oa-1-A.svg""></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.39.42 pm.png""></div>
</div>
<div id=""io-footer"">if it is a normal linear model</div>
<button id=""io-revl-btn"" onclick=""toggle();"">Toggle Masks</button>
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</div>
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<script>
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var amask = document.getElementById('io-overlay');
if (amask.style.display === 'block' || amask.style.display === '')
amask.style.display = 'none';
else
amask.style.display = 'block'
}
// Prevent original image from loading before mask
aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
function loaded() {
var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
"
"
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""fe5ba200c55b4e09800cd526480c5d30-oa-2-Q.svg"" /></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.39.42 pm.png"" /></div>
</div>
<div id=""io-footer"">if it is a normal linear model</div>
<script>
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aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
function loaded() {
var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
""
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""fe5ba200c55b4e09800cd526480c5d30-oa-2-A.svg""></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.39.42 pm.png"" /></div>
</div>
<div id=""io-footer"">if it is a normal linear model</div>
<button id=""io-revl-btn"" onclick=""toggle();"">Toggle Masks</button>
<div id=""io-extra-wrapper"">
<div id=""io-extra"">
</div>
</div>
<script>
// Toggle answer mask on clicking the image
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var amask = document.getElementById('io-overlay');
if (amask.style.display === 'block' || amask.style.display === '')
amask.style.display = 'none';
else
amask.style.display = 'block'
}
// Prevent original image from loading before mask
aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
function loaded() {
var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
"
"
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""44069a3454f04885a4af4c9ff09dbd0d-oa-1-Q.svg"" /></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.41.21 pm.png""></div>
</div>
<div id=""io-footer""></div>
<script>
// Prevent original image from loading before mask
aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
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var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
""
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""44069a3454f04885a4af4c9ff09dbd0d-oa-1-A.svg""></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.41.21 pm.png""></div>
</div>
<button id=""io-revl-btn"" onclick=""toggle();"">Toggle Masks</button>
<div id=""io-extra-wrapper"">
<div id=""io-extra"">
<div class=""io-extra-entry"">
<div class=""io-field-descr"">Extra 1</div><img src=""Screenshot 2024-03-17 at 1.41.35 pm.png"">
</div>
</div>
</div>
<script>
// Toggle answer mask on clicking the image
var toggle = function() {
var amask = document.getElementById('io-overlay');
if (amask.style.display === 'block' || amask.style.display === '')
amask.style.display = 'none';
else
amask.style.display = 'block'
}
// Prevent original image from loading before mask
aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
function loaded() {
var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
"
"
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""44069a3454f04885a4af4c9ff09dbd0d-oa-2-Q.svg"" /></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.41.21 pm.png"" /></div>
</div>
<div id=""io-footer""></div>
<script>
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var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
""
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""44069a3454f04885a4af4c9ff09dbd0d-oa-2-A.svg""></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.41.21 pm.png"" /></div>
</div>
<button id=""io-revl-btn"" onclick=""toggle();"">Toggle Masks</button>
<div id=""io-extra-wrapper"">
<div id=""io-extra"">
<div class=""io-extra-entry"">
<div class=""io-field-descr"">Extra 1</div><img src=""Screenshot 2024-03-17 at 1.41.35 pm.png"">
</div>
</div>
</div>
<script>
// Toggle answer mask on clicking the image
var toggle = function() {
var amask = document.getElementById('io-overlay');
if (amask.style.display === 'block' || amask.style.display === '')
amask.style.display = 'none';
else
amask.style.display = 'block'
}
// Prevent original image from loading before mask
aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
function loaded() {
var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
"
"
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""44069a3454f04885a4af4c9ff09dbd0d-oa-3-Q.svg""></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 1.41.21 pm.png"" /></div>
</div>
<div id=""io-footer""></div>
<script>
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aFade = 50, qFade = 0;
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var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
""
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""44069a3454f04885a4af4c9ff09dbd0d-oa-3-A.svg""></div>
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</div>
<button id=""io-revl-btn"" onclick=""toggle();"">Toggle Masks</button>
<div id=""io-extra-wrapper"">
<div id=""io-extra"">
<div class=""io-extra-entry"">
<div class=""io-field-descr"">Extra 1</div><img src=""Screenshot 2024-03-17 at 1.41.35 pm.png"">
</div>
</div>
</div>
<script>
// Toggle answer mask on clicking the image
var toggle = function() {
var amask = document.getElementById('io-overlay');
if (amask.style.display === 'block' || amask.style.display === '')
amask.style.display = 'none';
else
amask.style.display = 'block'
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aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
function loaded() {
var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
"
"
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""04cf18f73fa644ef8ed3587c08908a03-oa-1-Q.svg""></div>
<div id=""io-original""><img src=""Screenshot 2024-03-17 at 2.05.25 pm.png""></div>
</div>
<div id=""io-footer""></div>
<script>
// Prevent original image from loading before mask
aFade = 50, qFade = 0;
var mask = document.querySelector('#io-overlay>img');
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var original = document.querySelector('#io-original');
original.style.visibility = ""visible"";
}
if (mask === null || mask.complete) {
loaded();
} else {
mask.addEventListener('load', loaded);
}
</script>
""
<div id=""io-header""></div>
<div id=""io-wrapper"">
<div id=""io-overlay""><img src=""04cf18f73fa644ef8ed3587c08908a03-oa-1-A.svg""></div>
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""
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"
"<div class=mystyle1>statistical-modelling::inference</div>
<br>
What is the formula for the mean square due to regression MS_R"\(MS_R = \frac{SS_R}{p-1}\)
"
<div id=""io-header"">cochran's theorem</div>
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""
<div id=""io-header"">cochran's theorem</div>
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"
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""
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"
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""
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"
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""
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"
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""
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""
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"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
Formula for the coefficient of (multiple) determination&nbsp;\(R^2\)""<img src=""Screenshot 2024-03-17 at 2.36.05 pm.png"">"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What is considered a good value for&nbsp;\(R^2\)?&nbsp;"greater than 90% is considered good<br>greater than 80% is considered satisfactory<br><br>\(R^2\)&nbsp;alone does not tell you if the model is good or bad, only gives one aspect of the goodness of fit
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What is the formula for the standardised residual?&nbsp;""<img src=""Screenshot 2024-03-19 at 2.13.41 pm.png"">"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What is the distribution of standardised residuals for large n?&nbsp;""<img src=""Screenshot 2024-03-19 at 2.14.45 pm.png"">"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
Sketch the expected residual plot if there is constant variance""we should see a a random scatter on either side of&nbsp;\(r_i =0\)<br><img src=""Screenshot 2024-03-19 at 2.16.08 pm.png"">"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
Sketch the expected residual plot if the variance increases""should give a funnel shape,&nbsp; we should see a larger scatter at larger&nbsp;\(\hat y_i\)<br><img src=""Screenshot 2024-03-19 at 2.17.23 pm.png"">"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What does a normal probability plot for normal distribution look like?&nbsp;""should be a straight line&nbsp;<br><img src=""Screenshot 2024-03-19 at 2.21.08 pm.png"">"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What does a normal probability plot look like when there is positive skew?&nbsp;""<img src=""Screenshot 2024-03-19 at 2.21.56 pm.png"">"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What does a normal probability plot look like when there is high kurtosis (heavy tails) ?&nbsp;""<img src=""Screenshot 2024-03-19 at 2.22.45 pm.png"">"
"<div class=mystyle1>statistical-modelling::5-model-selection</div>
<br>
What can we do if we detect non-constant variance? (heteroskedascity)""can do a response transformation such as<br><img src=""Screenshot 2024-03-19 at 2.26.22 pm.png"">"
"<div class=mystyle1>statistical-modelling::5-model-selection</div>
<br>
What is the Box-Cox transformation? What is the purpose?&nbsp;""<img src=""Screenshot 2024-03-19 at 2.27.27 pm.png""><br>can be used to transform a model if Y is continuous and not normally distributed, and the variance is not stable<br><br>finds the \(\lambda\)&nbsp;which optimises the likelihood to transform the linear model into that&nbsp;<br>then we have&nbsp;\(y^\lambda = x\beta + \varepsilon\)<br>"
"<div class=mystyle1>statistical-modelling::5-model-selection</div>
<br>
What are three common mistakes when interpreting fitted models?&nbsp;""- drawing causal conclusions when the study only supports descriptive relationships<br>- extrapolition beyond range of data including hidden extrapolation such as<br><img src=""Screenshot 2024-03-19 at 2.32.49 pm.png""><br>- extrapolitions to different conditions to what the data was collected under"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What is an outlier?&nbsp;""an observation whose response does not fit the pattern of the rest of the data<br><img src=""Screenshot 2024-03-19 at 2.32.49 pm.png"" width=""627"">"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What does an outlier look like in a residual plot?&nbsp;""<img src=""Screenshot 2024-03-19 at 2.35.45 pm.png"">"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
Steps to carry out if we find an outlier (3)""- check whether the observation was misrecorded or
miscopied and if so correct it. <br>- check if the unit sampled is different in kind from the rest, if so remove it<br>- If it seems correctly recorded we should rerun the
analysis excluding the outlier. If the conclusions from the second analysis differ
substantially from the first one we should report both."
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What is the leverage?"\(h_{ii}\)<br>this is the same&nbsp;\(h_{ii}\)as the standardised residual<br>- indicates how much yi is contributing to estimating the model&nbsp;
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What is the mean leverage?&nbsp;"\(\frac pn\)
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What is considered a large enough leverage to be cause for concern?&nbsp;"\(h_{ii} &gt; \frac{2p}n\)is large enough to be cause for concern<br>\(h_{ii} &gt; \frac{3p}n\)&nbsp;is considered very large
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What is an influential point? How is influence measured?&nbsp;""a point with high leverage<br>- is measured using cooks distance<br>D_i = \frac{d_i^2}{}<br><img src=""Screenshot 2024-03-19 at 2.43.36 pm.png""><br><img src=""Screenshot 2024-03-19 at 2.43.43 pm.png""><br>depends on the residual and the leverage"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What is&nbsp;multicollinearity?&nbsp;""some columns of the X matrix are close to being
linear combinations of other columns so&nbsp;\(X^TX\)&nbsp;is nearly singular"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What are the problems with multicollinearity? (4)&nbsp;"- numerical procedures for inverting \(X^TX\) can fail or be inaccurate or unstable<br>- the estimates are sensitive to small changes in the data (eg. due to rounding)<br>- some estimators wil be highly correlated, so different models might fit the data equally well<br>- parameter estimators will have large variances
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
How can we detect multicollinearity?&nbsp;""by calculating the&nbsp;Variance inflation factor (VIF)<br>look out for VIF &gt; 10&nbsp;<br><img src=""Screenshot 2024-03-19 at 2.51.48 pm.png"">"
"<div class=mystyle1>statistical-modelling::1-introduction</div>
<br>
what is the principle of parsimony"using the simplest model that suits the purpose
"<div class=mystyle1>statistical-modelling::1-introduction</div>
<br>
How is the partial F-test statistic derived?&nbsp;<br><img src=""Screenshot 2024-03-19 at 3.01.05 pm.png"">""<img src=""Screenshot 2024-03-19 at 3.01.48 pm.png"">"
"<div class=mystyle1>statistical-modelling::5-model-selection</div>
<br>
Formula for adjusted&nbsp;\(R^2\)""<img src=""Screenshot 2024-03-19 at 3.04.00 pm.png"">"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
what is the pdf of the exponential family?&nbsp;""<img src=""Screenshot 2024-03-19 at 3.11.59 pm.png""><br>The parameter θ is called the natural or canonical parameter. The parameter φ
is usually assumed known. If it is unknown then it is often called the nuisance
parameter.<br><br>can be thought of as a likelihood resulting from a single observation y<br><img src=""Screenshot 2024-03-27 at 11.34.22 am.png"">"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What is the log-likelihood of a function in the exponential family with one observation?&nbsp;""essentially remove the exponential part&nbsp;<br><img src=""Screenshot 2024-03-27 at 11.55.56 am.png"">"
"
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<div class=""io-field-descr"">Remarks</div><img src=""Screenshot 2024-03-27 at 11.58.26 am.png""><br>do chain rule<br>interchange derivate and integral (since the dy does not depend of theta<br>integral of f_y = 1 and so the derivative = 0
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"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What is the expected value of Y if it is from the exponential family?&nbsp;""\(E(Y) = b' (\theta)\)<br><br><img src=""Screenshot 2024-03-27 at 12.13.45 pm.png"">"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What is the variance of Y if it is from the exponential family?&nbsp;""\(Var (Y) = b''(\theta)a(\phi)\)<br><img src=""Screenshot 2024-03-27 at 12.14.21 pm.png"">"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
Outline how you would show that the Poisson distribution is part of the exponential family""<img src=""Screenshot 2024-03-27 at 12.16.28 pm.png""><br><img src=""Screenshot 2024-03-27 at 12.16.42 pm.png""><br>θ = log λ, b(θ) = exp θ, a(φ) = 1 and c(y, φ) = − log y!.<br><img src=""Screenshot 2024-03-27 at 12.16.34 pm.png""><br><img src=""Screenshot 2024-03-27 at 12.16.34 pm.png""><br>"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What are the properties of Y for a generalised linear model (GLM)?&nbsp;"- Y_i's independent&nbsp;<br>- Y_i's are from the exponential family<br>- there exists a link function g(.) describing the relationship between E(Y) and the lienar predictor<br>ie. g(E(Y)) = \nu = \X \beta<br><br>which describes the relationship between Y_i&nbsp;and the linear predictor
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What is the link function g(.)?&nbsp;""<img src=""Screenshot 2024-03-27 at 12.54.35 pm.png""><br>linear predictor η<br>it is monotonic and differentiable with respect to&nbsp;\({\mu = x\beta}\)"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What is the key idea for the difference in linear relationships for linear regression and GLMs?&nbsp;"in linear regression, E(Y) and&nbsp;\(\beta\)&nbsp;have a linear relationship<br>in GLM, g(E(Y)) and&nbsp;\(\beta\)&nbsp;have a linear relationship<br><br>if we set the identity link g(x) = x for the GLM, then we get the linear regression model (linear regression is a type of GLM)&nbsp;&nbsp;
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
Outline how you would choose the link function g(.)<br>eg. for bernouilli""since it links from E(Y) which is limited in domain, to \(x\beta\)&nbsp;which is over the entrie real line,&nbsp;<br>1. we pick g(.) such that its range is the entire real line<br><img src=""Screenshot 2024-03-27 at 1.05.55 pm.png""><br>2. use the canonical link&nbsp;\(g(u) = \theta\)<br><img src=""Screenshot 2024-03-27 at 1.08.17 pm.png"">"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What is the canonical link function?&nbsp;""\(g(µ) ≡ b'^{−1}
(µ)\)<br>so if you let&nbsp;\(g(µ) ≡ \theta\)<br>Under the canonical link, the canonical
parameter is equal to the linear predictor<br><div>\(\eta = g(E(y)) = \theta\)<br></div>"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What is the score function? What is the purpose?&nbsp;""partial derivative of the log-likelihood estimator<br>- we want to get it to = 0 for all estimators<br><img src=""Screenshot 2024-03-27 at 1.19.33 pm.png""><br>is used to estimate&nbsp;\(\beta\)&nbsp;by finding the maximum log likelihood estimator<br><br><div>Note there is no close form solution and we need to use numerical algorithms (fisher scoring or newton raphson)&nbsp;</div>"
"
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"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What residuals do you use for the Generalised linear model (GLM)?&nbsp;""- Pearson residual&nbsp;<br><img src=""Screenshot 2024-03-27 at 1.55.03 pm.png""><br>- deviance residual<br><img src=""Screenshot 2024-03-27 at 1.55.57 pm.png"">"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What is the pearson residual?&nbsp;""is the residual divided by the standard deviation<br><img src=""Screenshot 2024-03-27 at 1.55.03 pm.png"">"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What is the deviance residual&nbsp;""<img src=""Screenshot 2024-03-27 at 1.55.57 pm.png"">"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models statistical-modelling::revision</div>
<br>
Outline the steps to carry out a partial F-test""Conduct ANOVA<br><div>If we split \(X = [X^{(1)}, X^{(2)}]\) and \(\beta = [\beta^{(1)}, \beta^{(2)}]^T\). We may consider</div>
<div>simultaneous tests for collections of parameters, i.e.,</div>
<div>\(H0 : \beta^{(2)}= 0\)</div><div>\(H1 : \beta^{(2)}\neq 0, \dim (\beta^{(2)})= r.\)</div>
<div>This can be formulated as a comparison between a simpler model and a more complex model.</div>
<div>Full Model: \(Y = X\beta + \varepsilon\)</div>
<div>Recuced Model: \(V = X^{(1)} \beta^{(1)}+ \varepsilon\)</div>
<div>(under the reduced model we assume \(\beta^{(2) }= 0\).)</div><div><br></div>(H0): All coefficients removed from the full model are zero. (reduced model)<br>(H1): At least one of the coefficients removed from the full model is non-zero.<br><img src=""Screenshot 2024-04-23 at 12.59.54 pm.png""><br>\(F \sim F(r, n-p)\)<br><br>If the&nbsp;<strong>p-value</strong>&nbsp;corresponding to the F test-statistic is below the significance level, we can&nbsp;<strong>reject the null hypothesis</strong>.<br><br>"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
What does the partial F-test compare?&nbsp;""<span style=""color: rgb(17, 17, 17); background-color: rgb(247, 247, 247);"">A&nbsp;</span><strong>partial F-test</strong><span style=""color: rgb(17, 17, 17); background-color: rgb(247, 247, 247);"">&nbsp;is used to determine whether there is a statistically significant difference between a&nbsp;</span><strong>full regression model</strong><span style=""color: rgb(17, 17, 17); background-color: rgb(247, 247, 247);"">&nbsp;and a&nbsp;</span><strong>nested version</strong><span style=""color: rgb(17, 17, 17); background-color: rgb(247, 247, 247);"">&nbsp;of the same model.</span>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<img src=""Screenshot 2024-03-27 at 2.24.44 pm.png"">"d. is not a normal linear model
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<img src=""Screenshot 2024-03-27 at 2.30.22 pm.png"">""<img src=""Screenshot 2024-03-27 at 2.30.38 pm.png""><br><img src=""paste-97fbecad6f8ee77e92b7d02e374d540371995314.jpg"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
True or false:&nbsp;<br><span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">To use a least squares estimator for the coefficients of a linear model, it is essential to assume that the errors are normally distributed.</span>"False<br><div>- The assumption of Gaussian errors is not essential to use the ordinary least squares estimator for the the model coefficients, which is valid also for a general linear model.</div>we know since its also valid for GLM which does not have normally distributed errors<br>
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">What do we do if X^TX</span><span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">&nbsp;is not invertible?</span>""<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">Try removing some basis functions/predictors<br></span>- since we know that some of the columns are linearly dependent, we need to try to remove them"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">What are some properties for the (ordinary) least squares estimator&nbsp;</span>\(\hat \beta\)<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">&nbsp;</span><span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">of the coefficients</span> \(\beta\)<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">&nbsp;of the general linear model?</span>""-&nbsp;\(\hat \beta\)<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">&nbsp;</span>is an unbiased estimator.<br><span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);""><b><i>-&nbsp;</i></b>&nbsp;</span>\(\hat \beta\)<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">&nbsp;</span>i<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">s the best linear unbiased estimator</span>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is&nbsp;\(Cov(\hat \beta)\)?"\(\sigma^2 (X^TX)^{-1}\)
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Formula to calculate the confidence interval of \(\beta_j\)"\(\hat \beta_j \pm t_{1-\frac \alpha 2, n-p} \sqrt {[\hat \sigma^2 (X^TX)^{-1}]_{jj})}\)
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Formula for the confidence interval of&nbsp;\(\hat \mu\)""<img src=""Screenshot 2024-03-27 at 3.14.41 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">The scatterplot of the residuals versus the fitted values is useful to check (3)</span>""<span style=""background-color: rgb(250, 252, 255);"">-&nbsp;</span><span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">if the variance of the error is constant<br>-&nbsp;</span><span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">if there is any trend in the residuals, thus some term is missing in the regression model.</span><span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);""><br></span>-&nbsp;<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">if there is any outlier</span>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<img src=""Screenshot 2024-03-27 at 5.26.07 pm.png"">"model 2<br>p value &lt; 0.05 so we reject null hypothesis that the betas can be set to 0<br>
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
When looking at AIC which model is preferrable?&nbsp;"model with lower AIC value<br><br><div>Akaike information criterion&nbsp;(AIC) is an estimator of prediction error&nbsp;</div>
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
How many observations are there in the dataset?&nbsp;<br><img src=""Screenshot 2024-04-01 at 1.50.37 pm.png"">"degrees of freedom RSE = n-p<br>p = number of \beta in the model which = 5 (one for each variable and one for intercept)<br>n = 7+5 = 12
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Analyse the results of a global F-test"H0 :&nbsp;\(\beta_1 = \beta_2 = ... = \beta_p = 0\)&nbsp;(the reduced model with only the intercept)&nbsp;<br>H1 = at least one of the&nbsp;\(\beta_n \neq 0\)&nbsp;<br><br>if p-value is less than the significanc level (0.05), we reject H0 and can conclude that Y seems to be related to at least one of the explanatory variables
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What does the R squared value mean? eg. R squared value = 0.94"means that 94% of the variation in Y is explained by the model
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Outline the steps to interpreting these results<br><img src=""Screenshot 2024-04-01 at 1.58.27 pm.png"">""<img src=""IMG_C12A3E69F87C-1.jpeg""><br>- global F-test<br>- individual t-tests do they give evidence that the explanatory variable is related to yield?&nbsp;<br>- describe the estimate for each variable and how this affects the yield value (eg. increasing heat increases yield by about 7.3)<br>- describe the r squared value (this percent of the variation is explained by the model)"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is the prediction interval of Y?&nbsp;"\(\hat Y_0 \pm t_{n-p,1- \frac \alpha 2} \sqrt {s^2(\underline x_0^T(X^TX)^{-1}\underline x_0+1)}\)&nbsp;<br>then can set&nbsp;\(\hat Y_0 = \underline x_0^T\underline {\hat \beta}\)
"<div class=mystyle1>statistical-modelling::4-model-checking statistical-modelling::revision</div>
<br>
Sketch a QQ plot with light tails""<img src=""Screenshot 2024-04-01 at 2.44.12 pm.png"">"
"<div class=mystyle1>statistical-modelling::5-model-selection statistical-modelling::revision</div>
<br>
How would you use Box-Cox transformation to choose which model to fit next&nbsp;<br><img src=""Screenshot 2024-04-01 at 2.54.00 pm.png"">""since \lambda is greatest at -1 and we use&nbsp;<br><img src=""Screenshot 2024-04-01 at 2.54.27 pm.png""><br>then we try&nbsp;<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">lm(1/y~x1+x2)&nbsp;<br></span><br>note: when we transform it we also include the residual term&nbsp;<br>we write&nbsp;\(y^{-1} = X \beta + varepsilon\)<br><img src=""Screenshot 2024-04-01 at 2.56.21 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is functional marginality""if any term in in the model, then any term marginal to it must also be in the model<br><img src=""Screenshot 2024-04-01 at 3.00.35 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">Which of the following quantities can be used to compare two or more models, not necessary nested?<br></span><div>a. The number of parameters.</div><div><div>b. AIC</div><i></i></div><div>c. 𝑠²</div><div><div>d. SSE</div></div><div><div>e. Adjusted&nbsp;<span style=""color: inherit;""></span>𝑅²</div><i></i></div><div><div>f. K-fold Cross-validation error</div><i></i></div>""b. AIC<br>e. Adjusted&nbsp;<span style=""color: inherit;""></span>𝑅^2<br>f. K-fold Cross-validation error"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is the difference between&nbsp;<br><span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">lm(y~x1*x2)</span><br><span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">lm(y~x1+x1:x2)<br>where x2 is categorical variable<br></span>which one should we use?&nbsp;""<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">lm(y~x1*x2) (since this has x1, x2, x1:x2<br></span>since we need to follow functional marginality and&nbsp;<span style=""color: rgb(46, 46, 46); background-color: rgb(250, 252, 255);"">lm(y~x1+x1:x2) does not include the term x2</span>"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models statistical-modelling::revision</div>
<br>
<img src=""Screenshot 2024-04-01 at 3.28.39 pm.png"">""<img src=""IMG_DD68E3D65631-1.jpeg"">"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
What are the disadvantages of adding high order polynomial terms to the model?&nbsp;""A polynomial of degree d − 1 can exactly fit any d points. <br>Using a high-order polynomial can lead to curves which come very
close to the data we used to estimate them, but predict very badly. (High
variances but small biases)<br>- leads to overfitting&nbsp;"
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
How can you avoid overfitting"use cross validation<br>part of the training data is used to test the data, high error rate indicates overfitting<br>so we are looking for a low RMSE&nbsp;
"<div class=mystyle1>statistical-modelling::4-model-checking</div>
<br>
H0 : 3β_1 − β_2 = 0, β_0 + β_3 = β_2<br><img src=""Screenshot 2024-04-10 at 3.36.12 pm.png""><br>Conduct the hypothesis test"The first null hypothesis H0 : 3β_1 − β_2 = 0 is not rejected (p = 0.244)<br>The second null hypothesis H0 : β_0 + β_3 = β_2 is not rejected (p = 0.223)
"<div class=mystyle1>statistical-modelling::5-model-selection</div>
<br>
What are the disadvantages of stepwise selection?&nbsp;""- “one-at-a-time” nature of adding/removing variables, means it is
possible to miss the “optimal” model&nbsp;<br>- The procedures are not directly linked to final objectives of prediction or
explanation <br>(may not really help solve the problem of interest. It is
important to keep in mind that model selection cannot be divorced from the
underlying purpose of the investigation)<br>- When there is an appreciable
degree of multicollinearity among the explanatory variables, the three
methods of selection may lead to quite different final models."
"<div class=mystyle1>statistical-modelling::5-model-selection</div>
<br>
What is Mallow's Cp? what are we looking for?&nbsp;""- compares the precision and bias of the full model to models with a subset of the predictors.<br>- smaller is better<br><span style=""color: rgb(71, 71, 71); background-color: rgb(255, 255, 255);"">look for models where&nbsp;</span><b>Mallows' Cp is small and close to the number of predictors in the model plus the constant (p)</b><span style=""color: rgb(71, 71, 71); background-color: rgb(255, 255, 255);"">. A small Mallows' Cp value indicates that the model is relatively precise</span>"
"<div class=mystyle1>statistical-modelling::5-model-selection</div>
<br>
What is the BIC? What are we looking for?&nbsp;"<div>&nbsp;Bayesian Information Criterion (BIC) is&nbsp;a measure of model fit that takes into account goodness of fit and the number of parameters used to achieve the fit.</div><div>- lower is better</div>
"<div class=mystyle1>statistical-modelling::5-model-selection</div>
<br>
<img src=""Screenshot 2024-04-10 at 4.22.08 pm.png"">""<img src=""IMG_167C359FDCF3-1.jpeg"">"
"<div class=mystyle1>statistical-modelling::1-introduction</div>
<br>
<img src=""Screenshot 2024-04-10 at 4.35.59 pm.png"">""<img src=""Screenshot 2024-04-10 at 4.36.22 pm.png""><br>since we have matrix&nbsp;<br><img src=""IMG_6512F6ADB834-1.jpeg"" width=""205""><br>"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
Carry out the hypothesis test for&nbsp;\(\beta_j\)&nbsp;for GLM""<img src=""Screenshot 2024-04-23 at 12.34.39 pm.png"">"
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
if the model is adequate, what distribution does the pearson residuals follow?&nbsp;"\(r^p_i \sim N(0,1)\)
"<div class=mystyle1>statistical-modelling::8-generalised-linear-models</div>
<br>
Formula for&nbsp;\(C_p\)"\(\frac{SS_E}{MS_E \text { of full model }}+ 2p -n\)
"<div class=mystyle1>statistical-modelling::1-introduction</div>
<br>
What is the MLE of&nbsp;\(\sigma^2\)(\(\hat \sigma^2\))"\(\hat \sigma^2 = \frac{SS_E}{n-p}\)
"<div class=mystyle1>statistical-modelling::1-introduction</div>
<br>
What is the variance of&nbsp;\(\chi^2(k)\)&nbsp;?&nbsp;"variance = 2k
"<div class=mystyle1>statistical-modelling::inference</div>
<br>
State Cochran's theorem""<img src=""Screenshot 2024-03-17 at 2.27.34 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What are the assumptions for \(\varepsilon\)&nbsp;for the matrix form&nbsp;\(\bf Y = X \beta + \varepsilon\)? (4)"- linearity<br>\(\bf E( \varepsilon) = 0 \implies E(Y|X) = X \beta\)<br>- constant variance&nbsp;\(Var (\varepsilon_i )= \sigma^2 \implies Var(Y|X_i) = \sigma^2\)<br>- independence&nbsp;<br>\(e_1,... e_n\)&nbsp;are independent \implies \(Y_1,..., Y_n\)&nbsp;are independent<br>- normality<br>\(\varepsilon \sim N_n(0, \sigma^2I) \implies \bf Y|X \sim N_n(X\beta, \sigma^2I)\)
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What is the intercept of&nbsp;\(Y = X\beta + \varepsilon\)?&nbsp;</div>"<div>Is \(E(Y| x = 0) = \beta_0\)</div>
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Formula for the unbiased estimator""<div><span style=""white-space: pre;"">\(\hat \sigma^2 = s^2 = \frac{\sum_{i=1}^n (\hat \varepsilon_i^2)}{n-p} = \frac{1}{n-p} (\bf Y-X \hat\beta)^T(Y-X \hat\beta)\)</span></div>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What are the assumptions for \(\hat \beta\)&nbsp;under the linear model? (3)"- unbiasedness<br>\(E_{Y|X}(\hat \beta) = \beta\)<br>- variance<br>\(Var(\hat \beta) = \sigma^2 (X^TX)^{-1}\)<br>- normality<br>\(\hat \beta \sim N_p (\beta, \sigma^2 (X^TX)^{-1})\)<br>\(\hat \beta\)&nbsp;is the best linear unbiased estimator (BLUE)
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
How do you estimate the parameter \beta?&nbsp;"use least squares estimation&nbsp;<br>to minimise the sum of squares of the residual error
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is the slope of the model?&nbsp;"the expected change in the mean of Y if x_j increases by one unit when holding other variables constant<br>\(E(Y|x_j + 1) - E(Y|x_j) = \beta_j, j= 1, ..., p - 1.\)
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is the sampling distribution?&nbsp;""<img src=""Screenshot 2024-03-17 at 1.39.42 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What are the assumptions necessary for the sampling distribution?&nbsp;""assuming normality<br><img src=""Screenshot 2024-03-17 at 1.39.42 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Outline the hypothesis test for \beta""we make the normality assumption about \varepsilon<br><div>\(H0 : \beta_j = 0\)<br></div>
<div>\(H1: \beta_j \neq 0\)<br></div><div><br></div><div>test statistic:</div><div>\(t_j = \frac{\hat \beta_j}{\sqrt{[\hat \sigma ^2 \bf{(X^TX)}^{-1}]_{jj}}}\)</div><div><span style=""white-space: pre;"">\([\hat \sigma ^2 (X^TX)^{-1}]_{jj}\)</span>&nbsp;is the jth diagonal term for the matrix<br></div><div><br></div><div>under assumption that H0 is true then \(T_j \sim t_{(n-p)}\)&nbsp;student t distribution</div><div>(use this since we have estimated <span style=""white-space: pre;"">variance)</span></div><div><br></div>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is the fitted value&nbsp;\(\hat y_i\)?"\(\hat y_i = \hat \beta_0 + \hat \beta_1x_{i,1}+ ...+\hat \beta_{p-1}x_{i, p-1}, i = 1,..., n\)
"For residual e,&nbsp;<br>\(\sum_{i=1}^n e_i\)&nbsp;= <span class=""cloze"" data-cloze=""0"" data-ordinal=""1"">[...]</span><br>
""For residual e,&nbsp;<br>\(\sum_{i=1}^n e_i\)&nbsp;= <span class=""cloze"" data-ordinal=""1"">0</span><br>
"
"For residual e,&nbsp;<br>\(\sum_{i=1}^n y_i\)&nbsp;= <span class=""cloze"" data-cloze=""&#x5C;&#x28;&#x5C;sum&#x5F;&#x7B;i&#x3D;1&#x7D;&#x5E;n&#x20;&#x5C;hat&#x20;y&#x5F;i&#x5C;&#x29;"" data-ordinal=""1"">[...]</span><br>
""For residual e,&nbsp;<br>\(\sum_{i=1}^n y_i\)&nbsp;= <span class=""cloze"" data-ordinal=""1"">\(\sum_{i=1}^n \hat y_i\)</span><br>
"
"For residual e,&nbsp;<br> \(\sum_{i=1}^n x_{ij}e_i\)&nbsp;= <span class=""cloze"" data-cloze=""0"" data-ordinal=""1"">[...]</span>&nbsp;\(\forall j = 1,...,p\)<br>
""For residual e,&nbsp;<br> \(\sum_{i=1}^n x_{ij}e_i\)&nbsp;= <span class=""cloze"" data-ordinal=""1"">0</span>&nbsp;\(\forall j = 1,...,p\)<br>
"
"For residual e,&nbsp;<br> \(\sum_{i=1}^n \hat y_{i}e_i\)&nbsp;= <span class=""cloze"" data-cloze=""0"" data-ordinal=""1"">[...]</span>&nbsp;<br>
""For residual e,&nbsp;<br> \(\sum_{i=1}^n \hat y_{i}e_i\)&nbsp;= <span class=""cloze"" data-ordinal=""1"">0</span>&nbsp;<br>
"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is the&nbsp;Residual Sum of Squares (SS_E) for linear models?&nbsp;"\(SSE = \sum _{i=1}^n e_i^2 = \sum _{i=1}^n (y_i - \hat y_i)^2\)
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is the&nbsp;Total Sum of Squares SS_Tfor linear models?&nbsp;"\(SST = \sum _{i=1}^n (y_i - \bar y_i)^2\)
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is the regression sum of squares for linear models? SS_R&nbsp;"\(SS_R = \sum _{i=1}^n (\hat y_i - \bar y_i)^2\)
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What does SST describe?&nbsp;</div>"<div>The overall observed variability in response y. Is decomposed into a component corresponding to the residual errors and a component corresponding to the regression</div>
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What does R^2 measure?&nbsp;</div>"<div>The global adequacy of predictors x of y</div>
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What does adjusted R^2 measure?&nbsp;</div>"<div>Is a measure of the global adequacy of prectors x of y while also acknowledging that SSE decreases in expectation as p increases</div>
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Outline how to carry out a global F -test"H0 : \(\beta_1 = \beta_2 = ... = \beta_p = 0\)&nbsp;(the reduced model with only the intercept)&nbsp;<br>H1 = at least one of the \(\beta_n \neq 0\)<br><br>test statistic&nbsp;<br>\(F = \frac{SS_R/(p-1)}{SS_E/(n-p)} = \frac{MSR}{MSE}\)<br>if H0 is true and under the assumption of normality then&nbsp;\(F \sim F(p-1, n-p)\)<br><br>if p-value is less than the significanc level (0.05), we reject H0 and can conclude that Y seems to be related to at least one of the explanatory variables
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Outline how to answer:&nbsp;<br>Is the model overall a good fit for the data?&nbsp;<br>given the r code summary"- look at the adjusted R-squared (want greater than 0.8)<br>- look at the p-value for the global f- test (want p-value &lt; 0.05 so at least one of the predictors is useful in explaining the variability)&nbsp;
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What are the residual plots used for model checking?&nbsp;</div>""<div>Residual vs index</div><div>e_i vs i</div>
<div>Residual vs random variable</div><div>e_i vs x_{ij}</div>
<div>Residual vs fitted value</div><div>ei vs \hat y_i</div>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What should we observe with the residual plots if model assumptions are correct?&nbsp;</div>"<div>should be 'patternless'&nbsp;</div><div>should not exhibit systematic patterns in either mean-level or variability&nbsp;</div><div>the residuals should form a horizontal 'band' around zero, with equal variability around zero everywhere.</div>
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>When do we use QQ plot?&nbsp;</div>"to check the normality assumptions
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What can you do if the variance is not stable and the data is not normal?&nbsp;</div>"<div>Can apply the box-cox transformation</div>
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What is the formula for outliers?&nbsp;</div>""<div>The same as standardised residuals</div>
<div>\(d_i = \frac{e_i}{\sqrt{[\hat \sigma^2(1-h_{ii})}} \approx N(0,1)\)<br></div>
<div>Where&nbsp;\(h_{ii}\)&nbsp;is the 4th diagonal term of H&nbsp;(hat matrix)</div>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
How can you identify outliers?&nbsp;"has a large standardised residual&nbsp;
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What values of Cook's distance implies that it is an influential point?&nbsp;</div>"if D_i &gt; 1, y_i is influential point
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>Formula for the partial F test statistic</div>""<img src=""Screenshot 2024-04-23 at 12.59.54 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>When do you use the general linear hypothesis?&nbsp;</div>""<div>If you want to test a more general type of constraint on the model</div>
<div>\(H0: \bf A \beta = 0\)<br></div>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What are the special types of predictors?&nbsp;"Polynomial terms<br>Factor/ categorical predictors: discrete predictors measured on a nominal (non-numerical) scale<br>interactions: an interaction is a modification of the effect of one predictor in the presence of another
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What are the stepwise model selection strategies? What is the goal?&nbsp;</div>""<div>Want to find the simplest possible model that adquately explains the observed response data</div>
Forward selection<br>backward elimination<br>stepwise selection<br>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What test do we use for model selection</div>""<div>We use partial f-test (we have one model nested in another model)&nbsp;</div>
Not univariate t-test&nbsp;<br>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>List the model selection criteria (5)</div>""<div>R^2, adjusted R^2</div>
<div>Mallow’s C_p</div>
<div>AIC</div>
<div>BIC<br>
cross validation</div>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>Why might the selection criteria be better than stepwise selection?&nbsp;</div>"<div>Dont need to look at nested models, can choose any of the predictors,&nbsp;</div><div>“one-at-a-time” nature of adding/removing variables, means it is possible to miss the “optimal” model for stepwise, but the selection criteria do not have this issue<br></div>
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>Why is LSE preferable over MLE?&nbsp;</div>""<div>LSE gives unbiased so is preferred,&nbsp;</div>
<div>MLE can give a biased estimator</div>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What is the general linear hypothesis useful for testing?&nbsp;</div>""to test for Lack of Fit when fitting a series of polynomial regression models to determine the appropriate degree of polynomial<span style=""color: rgb(31, 31, 31); background-color: rgb(255, 255, 255);"">.&nbsp;</span>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What is the canonical parameter of the exponential family</div>"\(\theta\)
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What is the dispersion parameter of the exponential family</div>"\(\phi\)
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Formula for the score function&nbsp;""<img src=""Screenshot 2024-04-23 at 12.32.29 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<div>What test do you use for model selections of GLM?&nbsp;</div>"<div>use likelihood ratio test (instead of partial f-test)</div><div>or AIC</div>
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Compute a 95% confidence
interval for the daily metal loss in environment A<br><img src=""Screenshot 2024-04-23 at 12.38.10 pm.png"">""<img src=""Screenshot 2024-04-23 at 12.43.21 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Comparing teacher salary against school size with:&nbsp;<br>size.coded=1 for small districts, size.coded=2 for median districts, and size.coded=3 for large district<br>Write the form of the fitted model and interpret the coefficients<br><img src=""Screenshot 2024-04-23 at 12.45.34 pm.png"">""<div>Based on R code and output, size. coded-1 (small size district) was used as baseline, and the dummies for size of district were created as following:</div><div><img src=""Screenshot 2024-04-23 at 12.48.47 pm.png""><br></div><div>\(\hat y = 31805.6 + 3520.5 × \text{size.f2} + 6028.5 × \text{size.f3}\)<br></div><div><br></div><div>B1 = 3520.5 represents the difference in average teachers salary between median and small school districts.<br>B2 = 6028.5 represents the difference in average teachers salary between large and small school districts<br></div>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Comparing teacher salary against school size with:&nbsp;<br>size.coded=1 for small districts, size.coded=2 for median districts, and size.coded=3 for large district<br><img src=""Screenshot 2024-04-23 at 12.49.53 pm.png""><br>how would you&nbsp;test whether there is a difference
in teacher’s salary between median and large districts?""We want to test the hypothesis<br>
H0 : B1 - B2 = 0 v.s. H1 : B1 - B2 \neq 0<br>Use the general linear hypothesis, with A = [0, 1, -1]."
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
fit 2:&nbsp;lm(formula = salary ~ size.f + experience, data = iowastudent)<br><img src=""Screenshot 2024-04-23 at 12.53.57 pm.png""><br>fit3&lt;-lm(salary~size.f + experience+size.f*experience, data=iowastudent)<br><img src=""Screenshot 2024-04-23 at 12.54.18 pm.png""><br>how can you test to see if the relationship between teachers salary and experience
depends on the school district size?""<p class=""p1"">To test whether interaction terms are significant in fit3, i.e.,</p>
<p class=""p1"">H0: \beta_4 = \beta_5 = 0 v.s. H1: at least one is not 0</p><p class=""p1"">we use the partial F -test and the statistic is&nbsp;</p><p class=""p1""><img src=""Screenshot 2024-04-23 at 1.09.16 pm.png""><br></p>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is the formula for residual standard error?&nbsp;"\(RMSE = \sqrt{\frac{SSE}{n-p}}\)
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Formula for the SSE given the r-code summary&nbsp;"SSE = (Residual standard error)² x (n-p)<br>n-p is the degrees of freedom
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is the AIC of fit2?&nbsp;<img src=""Screenshot 2024-04-23 at 1.11.02 pm.png"">"513.84
"<div class=mystyle1>statistical-modelling::4-model-checking statistical-modelling::revision</div>
<br>
Sketch the residual plot if the true model is actually&nbsp;\(Y^2_i = x^T_i β + \epsilon_i\)""<img src=""Screenshot 2024-05-03 at 2.37.27 pm.png""><br><img src=""paste-1d07ca17f4dc8f84d13530719a332c736bf34fe7.jpg"">"
"<div class=mystyle1>statistical-modelling::4-model-checking statistical-modelling::revision</div>
<br>
Sketch the residual plot if the true error distribution is&nbsp;\(\epsilon_i \sim t_3\)""<span style=""background-color: rgb(255, 255, 255);"">qqplot&nbsp;has&nbsp;inverted&nbsp;S&nbsp;shape&nbsp;<br></span><img src=""Screenshot 2024-05-03 at 2.40.11 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Explain what these plots are used for. What do they tell us in this case?<br><img src=""Screenshot 2024-05-03 at 2.50.34 pm.png"">""The first plot allows us to check for heteroskedasticity,<br>- is the variance of the residual constant?&nbsp;<br>- any notable outliers?&nbsp;<br>- any underlying patterns indicating missing terms?&nbsp;<br><br>the second for
non-normality and also outliers<br><br>There are no major problems here."
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Given that we have the model<br><img src=""Screenshot 2024-05-03 at 2.54.31 pm.png""><br>Find the least squares estimate of the parameters of the model""\(\hat\beta=(X^TX)^{-1}X^TY\)<span style=""background-color: rgb(255, 255, 255);"">&nbsp;&nbsp;</span><img src=""Screenshot 2024-05-03 at 2.55.11 pm.png""><img src=""Screenshot 2024-05-03 at 2.55.17 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Explain how you would use the fitted model to make a recommendation on how x affects y<br>where x is categorical data""<span style=""background-color: rgb(255, 255, 255);"">-&nbsp;test&nbsp;interactions&nbsp;to&nbsp;see&nbsp;if&nbsp;significant&nbsp;<br>-&nbsp;estimate&nbsp;expected&nbsp;responses&nbsp;for&nbsp;each&nbsp;combination&nbsp;and&nbsp;recommend&nbsp;the&nbsp;one&nbsp;which&nbsp;gives&nbsp;the&nbsp;most&nbsp;desired&nbsp;y&nbsp;value</span>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is&nbsp;\((X^TX)^{-T}\)?"\((X^TX)^{-1}\)
"<div class=mystyle1>statistical-modelling::revision</div>
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What is&nbsp;\(\text{Var}(\textbf{AX})\)&nbsp;where X is matrix?&nbsp;"\(A\text{Var}(\textbf{X}) A^T\)
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"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
State the confidence interval for error variance""<img src=""IMG_FECAE15FDBF7-1.jpeg"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
<img src=""Screenshot 2024-05-05 at 3.33.25 pm.png"">""<div>Yes, because the difference in deviance between the null model and the model with age and the quadratic term is&nbsp;51.817-44.673=7.144, which is larger than the&nbsp;95%quantile of the<span style=""white-space: pre;"">&nbsp;\(𝜒^2_2(≈5.99).\)</span></div>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Formula for Cooks distance""<img src=""Screenshot 2024-03-19 at 2.43.36 pm.png""><br><img src=""Screenshot 2024-03-19 at 2.43.43 pm.png""><br><img src=""Screenshot 2024-05-08 at 1.43.42 pm.png""><br>where d_i is the standardised residual and p is number of coefficients in model<br>s^2 = MSE"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Outline how to carry out a hypothesis test for model 1 against model 2 for likelihood ratio test<br>model 1:&nbsp;<br><img src=""Screenshot 2024-05-08 at 2.01.27 pm.png""><br><br>model 2:<br><img src=""Screenshot 2024-05-08 at 4.13.15 pm.png"">""<img src=""Screenshot 2024-05-08 at 2.01.46 pm.png""><br><img src=""Screenshot 2024-05-08 at 2.01.56 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
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Analysis of deviance table<br><img src=""Screenshot 2024-05-08 at 2.24.01 pm.png""><br>Summarise the output""We consider to test that the year of operation is unimportant. Based on LRTs,
we reject H0 and conclude that the accident rate across ships of different periods
of operation are different"
"<div class=mystyle1>statistical-modelling::revision</div>
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What is the odds ratio?"\(\frac{\pi_i}{1-\pi_i}\)<br>odd of event vs no event
"<div class=mystyle1>statistical-modelling::revision</div>
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What is the link for binomial distribution?"logit link<br>\(g(\pi) = \log \frac{\pi}{1-\pi}\)
"<div class=mystyle1>statistical-modelling::revision</div>
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What is the complementary log log link?&nbsp;"\(g(\pi) = \log(-\log(\pi))\)
"<div class=mystyle1>statistical-modelling::revision</div>
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what is \(\beta_1\)&nbsp;in terms of the odds ratio for binary regression?&nbsp;<br><img src=""IMG_88B025FD67FD-1.jpeg"">""<img src=""IMG_A199FB8A75F7-1.jpeg"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What can we say when the odds ratio is 1?&nbsp;"the odds of event in group 2 are the same as the odds of event in group 1 which indicates that x is not associated with the event
"<div class=mystyle1>statistical-modelling::revision</div>
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What can we say if the odds ratio is greater than 1?&nbsp;"then the event is more likely to happen in group 2 (the one associated with the ratio on top)&nbsp;
"<div class=mystyle1>statistical-modelling::revision</div>
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Formula for the confidence interval of odds ratio using \(\hat \beta_j\)&nbsp;for GLM on binomial regression""need to find the confidence interval and then use the exponential to reverse the log link<br><img src=""Screenshot 2024-05-08 at 3.01.41 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
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using binary regression<br><img src=""Screenshot 2024-05-08 at 3.03.50 pm.png"">""<img src=""Screenshot 2024-05-08 at 3.03.55 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Calculate the confidence interval of of the odds ratio using&nbsp;\(\hat \beta_2 - \hat \beta_1\)&nbsp;for binary regression<br><img src=""Screenshot 2024-05-08 at 3.07.20 pm.png"">""We use the fisher information table to obtain the variance-covariance matrix<br><img src=""Screenshot 2024-05-08 at 3.08.09 pm.png""><br><img src=""Screenshot 2024-05-08 at 3.08.18 pm.png""><br><img src=""Screenshot 2024-05-08 at 3.09.46 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
How would you calculate the sensitivity of this prediction?&nbsp;<br><img src=""Screenshot 2024-05-08 at 3.17.48 pm.png"">""<img src=""Screenshot 2024-05-08 at 3.18.35 pm.png""><br><img src=""Screenshot 2024-05-08 at 3.18.55 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
How would you calculate the specificity of this prediction?&nbsp;<br><img src=""Screenshot 2024-05-08 at 3.17.48 pm.png"">""<img src=""Screenshot 2024-05-08 at 3.18.44 pm.png""><br><img src=""Screenshot 2024-05-08 at 3.18.57 pm.png"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What is the&nbsp;Receiver operating characteristic curve? What are we aiming for?&nbsp;<br><img src=""Screenshot 2024-05-08 at 3.21.38 pm.png"">""<img src=""Screenshot 2024-05-08 at 3.21.44 pm.png""><br>we want as large an area under the curve as possible.<br>over 0.8 is okay<br>"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
What distribution do we follow for count data?&nbsp;""Poisson distribution<br>- canonical link = log link<br><img src=""IMG_27B7BF41AB8C-1.jpeg"">"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Find the relative
rate of accidents for ships of type A versus type C.<br><img src=""Screenshot 2024-05-08 at 3.48.07 pm.png"">""<img src=""Screenshot 2024-05-08 at 3.49.08 pm.png""><br>For ships constructed in the same period and operated in the same period
the rate of accidents for ships of type A is 1.99 times higher than the rate
of accidents for ships of type C."
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Find the 95% confidence interval for MLE based on the asymptotic normality of the MLE when using poisson regression""<img src=""Screenshot 2024-05-08 at 3.50.39 pm.png""><br>can find all the data on the summary table<br>where&nbsp;\(\hat\beta_2\)&nbsp;is the log likelihood<br>(is why we need to exponentiate)&nbsp;"
"<div class=mystyle1>statistical-modelling::revision</div>
<br>
Find the relative rate of accidents for ships of type B versus type E.<br><img src=""Screenshot 2024-05-08 at 3.48.07 pm.png"">""<img src=""Screenshot 2024-05-08 at 3.56.01 pm.png"">"