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body { font-family: Segoe UI; font-size: 12pt; } </style></head><body><div><br /></div><div><br /></div><div>Dear Michel, </div><div><br /></div><div><div id="xbcf472f85b1d4823974254d3d958088b">
<p class="MsoNormal" style="margin-top:3.25pt;line-height:12.8pt;mso-line-height-rule:
exactly;vertical-align:baseline" xmlns="http://www.w3.org/TR/REC-html40"><a name="_GoBack"></a><span style="font-size: 12pt; letter-spacing: -0.2pt;">Any residual with an absolute value which is equal to or greater than
the tabled critical two-tailed .05 value <b><i>z </i></b><sub>05</sub> = 1.96 is
significant at the .05 level. Any residual which is equal to or greater than
the tabled critical two-tailed .01 <b><i>z </i></b><sub>01</sub> = 2.58 is
significant at the .01 level. Any cell in a contingency table that has a significant
residual makes a significant contribution to the obtained chi-square value. For
any cell whit a significant residual, one can conclude that the observed
frequency of the cell differs
significantly from its expected frequency. The sign of the standardized
residual indicates whether the observed frequency of the cell is above (+) or
below (-) the expected frequency. The sum of the squared residuals for all <i>r x c </i>cells will equal the obtained
value of chi-square. <o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></span></p>
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</p><div>
<span style="font-family: Arial; background-color: rgb(255, 255, 255);">Sheskin, D. J. (2011). </span>
<i>
<span style="font-family: Arial; background-color: rgb(255, 255, 255);">Handbook of Parametric and Nonparametric Statistical Procedures (5th Edition)</span>
</i>
<span style="font-family: Arial; background-color: rgb(255, 255, 255);">. Boca Raton, FL: Chapman & Hall/CRC. p. 672.</span></div><p class="MsoNormal" xmlns="http://www.w3.org/TR/REC-html40"><span style="font-size:12.0pt"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"><br /></o:p></span></p></div></div>
<div><br /></div><div id="x491ea7b2763a40a"><blockquote cite="CAEw5oCbG3=zi+exsv5c4r4S4z3sa4dXM=eTFsP8o+U+Gx8z_Ng@mail.gmail.com" type="cite" class="cite2"><div dir="ltr"><div dir="ltr"><div class="gmail_default" style="font-family:monospace,monospace">I doubt that the square root of (observed - expected) makes sense: is the quantity inside parentheses always non-negative?</div></div></div></blockquote><font face="monospace, monospace">The sign is meaningful: above or below expectation?<br /></font><blockquote cite="CAEw5oCbG3=zi+exsv5c4r4S4z3sa4dXM=eTFsP8o+U+Gx8z_Ng@mail.gmail.com" type="cite" class="cite2"><div dir="ltr"><div dir="ltr"><div class="gmail_default" style="font-family:monospace,monospace"><br /></div><div class="gmail_default" style="font-family:monospace,monospace">May be you meant a chi-2 test on a 2x2 contingency table</div><div class="gmail_default" style="font-family:monospace,monospace"><a href="http://www.stat.yale.edu/Courses/1997-98/101/chisq.htm">http://www.stat.yale.edu/Courses/1997-98/101/chisq.htm</a></div><div class="gmail_default" style="font-family:monospace,monospace"><br /></div></div></div></blockquote><font face="monospace, monospace">You are right: the formulas were not correctly typed. However, the computation and interpretation was correct. </font></div><div id="x491ea7b2763a40a"><font face="monospace, monospace">The mail should have been without these typos:</font><br /></div><div id="x491ea7b2763a40a"><font face="monospace, monospace"><br /></font></div><div id="x491ea7b2763a40a"><blockquote cite="CAEw5oCbG3=zi+exsv5c4r4S4z3sa4dXM=eTFsP8o+U+Gx8z_Ng@mail.gmail.com" type="cite" class="cite2"><div dir="ltr"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div><div id="gmail-m_1877032139338450650xca030115c06040e">On second thought, it seems to me that a simple and most appropriate measure for the difference that one difference makes for another could be chi-square statistics.</div><div id="gmail-m_1877032139338450650xca030115c06040e"><br /></div><div id="gmail-m_1877032139338450650xca030115c06040e">For example, one can ask whether a specific treatment makes a difference differently for man and women. The two variables can be cross-tabled as in Table 1.</div><div id="gmail-m_1877032139338450650xca030115c06040e"><br /><div id="gmail-m_1877032139338450650x6e34d26becf445d4ad84c1af0aceb2c0">
<table style="width:218pt" width="292" cellspacing="0" cellpadding="0" border="0">
<colgroup><col style="width:47pt" width="63" />
<col style="width:61pt" width="82" />
<col style="width:63pt" width="84" />
<col style="width:47pt" width="63" />
</colgroup><tbody><tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;width:47pt" width="63" height="19">Observed</td>
<td class="gmail-m_1877032139338450650xl65" style="border-left:medium none;width:61pt" width="82">Treatment </td>
<td class="gmail-m_1877032139338450650xl65" style="border-left:medium none;width:63pt" width="84">No treatment</td>
<td style="width:47pt" width="63"></td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;border-top:medium none" height="19">male</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">156</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">63</td>
<td align="right">219</td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;border-top:medium none" height="19">female</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">86</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">74</td>
<td align="right">160</td>
</tr>
<tr style="height:14pt" height="19">
<td style="height:14pt" height="19"></td>
<td align="right">242</td>
<td align="right">137</td>
<td align="right">379</td>
</tr>
</tbody></table><br /></div><div id="gmail-m_1877032139338450650x6e34d26becf445d4ad84c1af0aceb2c0">This table with observed values may make a difference with reference to the expected values. The latter are determined using the margin totals:</div><div id="gmail-m_1877032139338450650x6e34d26becf445d4ad84c1af0aceb2c0"><br /></div><div id="gmail-m_1877032139338450650x6e34d26becf445d4ad84c1af0aceb2c0"><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1">
<table style="width:218pt" width="292" cellspacing="0" cellpadding="0" border="0">
<colgroup><col style="width:47pt" width="63" />
<col style="width:61pt" width="82" />
<col style="width:63pt" width="84" />
<col style="width:47pt" width="63" />
</colgroup><tbody><tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;width:47pt" width="63" height="19">Expected</td>
<td class="gmail-m_1877032139338450650xl65" style="border-left:medium none;width:61pt" width="82">Treatment </td>
<td class="gmail-m_1877032139338450650xl65" style="border-left:medium none;width:63pt" width="84">No treatment</td>
<td class="gmail-m_1877032139338450650xl66" style="width:47pt" width="63"></td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;border-top:medium none" height="19">male</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">140</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">79</td>
<td class="gmail-m_1877032139338450650xl66" align="right">219</td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;border-top:medium none" height="19">female</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">102</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">58</td>
<td class="gmail-m_1877032139338450650xl66" align="right">160</td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl66" style="height:14pt" height="19"></td>
<td class="gmail-m_1877032139338450650xl66" align="right">242</td>
<td class="gmail-m_1877032139338450650xl66" align="right">137</td>
<td class="gmail-m_1877032139338450650xl66" align="right">379</td>
</tr>
</tbody></table><br /></div><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1">for example for the first cell: 219 * 242/ 379 = 140.</div><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1"><br /></div><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1">Each corresponding cell contributes to the chi-square with the so-called standardized residual to the chi-square: </div><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1"><br /></div></div></div></div></blockquote></div></div></blockquote>
<div>
<img src="cid:emd639165f-bcdd-4b6d-a016-2c849fa70896@pc2014" width="256" height="61" />
</div><blockquote cite="CAEw5oCbG3=zi+exsv5c4r4S4z3sa4dXM=eTFsP8o+U+Gx8z_Ng@mail.gmail.com" type="cite" class="cite2"><div dir="ltr"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div><div id="gmail-m_1877032139338450650xca030115c06040e"><div id="gmail-m_1877032139338450650x6e34d26becf445d4ad84c1af0aceb2c0"><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1"><div>The residual of the chi-square is the square root of this:</div></div></div></div></div></blockquote><div class="gmail_quote"><br /></div></div></div></blockquote>
<div>
<img src="cid:em99ff62be-76b7-47d9-aa7b-5c6566651a96@pc2014" width="230" height="65" />
</div><blockquote cite="CAEw5oCbG3=zi+exsv5c4r4S4z3sa4dXM=eTFsP8o+U+Gx8z_Ng@mail.gmail.com" type="cite" class="cite2"><div dir="ltr"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div><div><br /></div><div>
<div> This is a <i>z-</i>statistic. For the four cells of the matrix, we can derive:</div><div><div id="gmail-m_1877032139338450650xca030115c06040e"><div id="gmail-m_1877032139338450650x6e34d26becf445d4ad84c1af0aceb2c0"><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1"><div><div></div><div></div></div></div></div></div></div><div id="gmail-m_1877032139338450650x092434b26b0d4b3e840a7b5fd0361586">
<table style="width:171pt" width="229" cellspacing="0" cellpadding="0" border="0">
<colgroup><col style="width:47pt" width="63" />
<col style="width:61pt" width="82" />
<col style="width:63pt" width="84" />
</colgroup><tbody><tr style="height:14pt" height="19">
<td style="height:14pt;width:47pt" width="63" height="19"></td>
<td style="width:61pt" width="82">Treatment</td>
<td style="width:63pt" width="84">No Treatment</td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt" height="19">male</td>
<td class="gmail-m_1877032139338450650xl66" style="border-left:medium none" align="right">-1.37</td>
<td class="gmail-m_1877032139338450650xl66" style="border-left:medium none" align="right">1.82</td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;border-top:medium none" height="19">female</td>
<td class="gmail-m_1877032139338450650xl66" style="border-top:medium none;border-left:medium none" align="right">1.60</td>
<td class="gmail-m_1877032139338450650xl67" style="border-top:medium none;border-left:medium none" align="right">-2.13</td>
</tr>
</tbody></table></div><div> </div></div><div>The difference between expected and observed is only significant (p <.05) for women without treatment:<i> z </i>= - 2.13. The threshold is 1.96 for the 5% level. The z-values can directly be used for the measurement of Bateson's information : "a difference that makes a difference. The difference of receiving treatment makes a difference for all four categories, but this difference is in tis case only significant for women without treatment. </div><div><br /></div><div>Thus, Bateson-type information can be measured. There is also a link to Shannon-type information via the log-likelyhood chi-square (which is a Shannon-type measure.)</div><div><br /></div><div>Problems solved? One can measure the value of the difference that makes a difference.</div><div><br /></div><div>Best,</div><div>Loet</div><div><br /></div></div>_______________________________________________<br />
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