document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170647>Question 1170647</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dr. James has initiated work on cancer systems biology. He has obtained 3 different cell lines for the expression (in nM) of five proteins. Here is the data that he got. <BR>\n" );
document.write( "Proteins\r<BR>\n" );
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document.write( "Cell lines	P53	Akt	Cyclin D	  mTor	       GLUT<BR>\n" );
document.write( "Colon	        130	121	160	         131             171<BR>\n" );
document.write( "Pancreatic	123	113	158	         106	          165<BR>\n" );
document.write( "Lung	         121	112	164	         102	            149 \r<BR>\n" );
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document.write( "At the 0.01 and 0.05 level of significance, is there a significant difference protein expression among?<BR>\n" );
document.write( " (a)Proteins? (b) Cell lines?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851124 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851124\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To determine if there's a significant difference in protein expression among proteins and cell lines, we'll use a two-way ANOVA (Analysis of Variance).\r<BR>\n" );
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document.write( "**1. Set up the Data**\r<BR>\n" );
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document.write( "Here's the data organized:\r<BR>\n" );
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document.write( "| Cell Line | P53 | Akt | Cyclin D | mTor | GLUT |<BR>\n" );
document.write( "|---|---|---|---|---|---|<BR>\n" );
document.write( "| Colon | 130 | 121 | 160 | 131 | 171 |<BR>\n" );
document.write( "| Pancreatic | 123 | 113 | 158 | 106 | 165 |<BR>\n" );
document.write( "| Lung | 121 | 112 | 164 | 102 | 149 |\r<BR>\n" );
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document.write( "**2. Calculate the Necessary Sums and Means**\r<BR>\n" );
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document.write( "* **Overall Mean (Grand Mean):**<BR>\n" );
document.write( "    * Sum of all values = 130 + 121 + 160 + ... + 149 = 1784<BR>\n" );
document.write( "    * Total number of values (N) = 15<BR>\n" );
document.write( "    * Grand Mean = 1784 / 15 ≈ 118.93\r<BR>\n" );
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document.write( "* **Cell Line Means:**<BR>\n" );
document.write( "    * Colon Mean = (130 + 121 + 160 + 131 + 171) / 5 = 142.6<BR>\n" );
document.write( "    * Pancreatic Mean = (123 + 113 + 158 + 106 + 165) / 5 = 133<BR>\n" );
document.write( "    * Lung Mean = (121 + 112 + 164 + 102 + 149) / 5 = 129.6\r<BR>\n" );
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document.write( "* **Protein Means:**<BR>\n" );
document.write( "    * P53 Mean = (130 + 123 + 121) / 3 = 124.67<BR>\n" );
document.write( "    * Akt Mean = (121 + 113 + 112) / 3 = 115.33<BR>\n" );
document.write( "    * Cyclin D Mean = (160 + 158 + 164) / 3 = 160.67<BR>\n" );
document.write( "    * mTor Mean = (131 + 106 + 102) / 3 = 113<BR>\n" );
document.write( "    * GLUT Mean = (171 + 165 + 149) / 3 = 161.67\r<BR>\n" );
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document.write( "**3. Calculate Sum of Squares (SS)**\r<BR>\n" );
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document.write( "* **SS Total (SST):**<BR>\n" );
document.write( "    * SST = Σ(xᵢ - Grand Mean)²<BR>\n" );
document.write( "    * SST = (130 - 118.93)² + (121 - 118.93)² + ... + (149 - 118.93)² ≈ 4165.73\r<BR>\n" );
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document.write( "* **SS Between Cell Lines (SS Cells):**<BR>\n" );
document.write( "    * SS Cells = Σn(Cell Mean - Grand Mean)² (where n is the number of proteins per cell line, which is 5)<BR>\n" );
document.write( "    * SS Cells = 5 * [(142.6 - 118.93)² + (133 - 118.93)² + (129.6 - 118.93)²] ≈ 3208.93\r<BR>\n" );
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document.write( "* **SS Between Proteins (SS Proteins):**<BR>\n" );
document.write( "    * SS Proteins = Σn(Protein Mean - Grand Mean)² (where n is the number of cell lines per protein, which is 3)<BR>\n" );
document.write( "    * SS Proteins = 3 * [(124.67 - 118.93)² + (115.33 - 118.93)² + (160.67 - 118.93)² + (113 - 118.93)² + (161.67 - 118.93)²] ≈ 893.33\r<BR>\n" );
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document.write( "* **SS Error (SSE):**<BR>\n" );
document.write( "    * SSE = SST - SS Cells - SS Proteins<BR>\n" );
document.write( "    * SSE = 4165.73 - 3208.93 - 893.33 ≈ 63.47\r<BR>\n" );
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document.write( "**4. Calculate Degrees of Freedom (df)**\r<BR>\n" );
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document.write( "* df Cells = Number of cell lines - 1 = 3 - 1 = 2<BR>\n" );
document.write( "* df Proteins = Number of proteins - 1 = 5 - 1 = 4<BR>\n" );
document.write( "* df Error = (Number of cell lines - 1) * (Number of proteins - 1) = 2 * 4 = 8<BR>\n" );
document.write( "* df Total = Total number of values - 1 = 15 - 1 = 14\r<BR>\n" );
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document.write( "**5. Calculate Mean Squares (MS)**\r<BR>\n" );
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document.write( "* MS Cells = SS Cells / df Cells = 3208.93 / 2 ≈ 1604.47<BR>\n" );
document.write( "* MS Proteins = SS Proteins / df Proteins = 893.33 / 4 ≈ 223.33<BR>\n" );
document.write( "* MS Error = SSE / df Error = 63.47 / 8 ≈ 7.93\r<BR>\n" );
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document.write( "**6. Calculate F-statistics**\r<BR>\n" );
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document.write( "* F Cells = MS Cells / MS Error = 1604.47 / 7.93 ≈ 202.33<BR>\n" );
document.write( "* F Proteins = MS Proteins / MS Error = 223.33 / 7.93 ≈ 28.16\r<BR>\n" );
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document.write( "**7. Determine Critical F-values**\r<BR>\n" );
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document.write( "* **α = 0.05:**<BR>\n" );
document.write( "    * F_critical (2, 8) ≈ 4.46<BR>\n" );
document.write( "    * F_critical (4, 8) ≈ 3.84<BR>\n" );
document.write( "* **α = 0.01:**<BR>\n" );
document.write( "    * F_critical (2, 8) ≈ 8.65<BR>\n" );
document.write( "    * F_critical (4, 8) ≈ 7.01\r<BR>\n" );
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document.write( "**8. Make Decisions**\r<BR>\n" );
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document.write( "**(a) Proteins**\r<BR>\n" );
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document.write( "* F Proteins (28.16) > F_critical (4, 8) at both 0.05 and 0.01.<BR>\n" );
document.write( "* **Conclusion:** There is a significant difference in protein expression among the proteins.\r<BR>\n" );
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document.write( "**(b) Cell Lines**\r<BR>\n" );
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document.write( "* F Cells (202.33) > F_critical (2, 8) at both 0.05 and 0.01.<BR>\n" );
document.write( "* **Conclusion:** There is a significant difference in protein expression among the cell lines.<BR>\n" );
document.write( " </SPAN>\n" );
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