Question 1170647: 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.
Proteins
Cell lines P53 Akt Cyclin D mTor GLUT
Colon 130 121 160 131 171
Pancreatic 123 113 158 106 165
Lung 121 112 164 102 149
At the 0.01 and 0.05 level of significance, is there a significant difference protein expression among?
(a)Proteins? (b) Cell lines?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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).
**1. Set up the Data**
Here's the data organized:
| Cell Line | P53 | Akt | Cyclin D | mTor | GLUT |
|---|---|---|---|---|---|
| Colon | 130 | 121 | 160 | 131 | 171 |
| Pancreatic | 123 | 113 | 158 | 106 | 165 |
| Lung | 121 | 112 | 164 | 102 | 149 |
**2. Calculate the Necessary Sums and Means**
* **Overall Mean (Grand Mean):**
* Sum of all values = 130 + 121 + 160 + ... + 149 = 1784
* Total number of values (N) = 15
* Grand Mean = 1784 / 15 ≈ 118.93
* **Cell Line Means:**
* Colon Mean = (130 + 121 + 160 + 131 + 171) / 5 = 142.6
* Pancreatic Mean = (123 + 113 + 158 + 106 + 165) / 5 = 133
* Lung Mean = (121 + 112 + 164 + 102 + 149) / 5 = 129.6
* **Protein Means:**
* P53 Mean = (130 + 123 + 121) / 3 = 124.67
* Akt Mean = (121 + 113 + 112) / 3 = 115.33
* Cyclin D Mean = (160 + 158 + 164) / 3 = 160.67
* mTor Mean = (131 + 106 + 102) / 3 = 113
* GLUT Mean = (171 + 165 + 149) / 3 = 161.67
**3. Calculate Sum of Squares (SS)**
* **SS Total (SST):**
* SST = Σ(xᵢ - Grand Mean)²
* SST = (130 - 118.93)² + (121 - 118.93)² + ... + (149 - 118.93)² ≈ 4165.73
* **SS Between Cell Lines (SS Cells):**
* SS Cells = Σn(Cell Mean - Grand Mean)² (where n is the number of proteins per cell line, which is 5)
* SS Cells = 5 * [(142.6 - 118.93)² + (133 - 118.93)² + (129.6 - 118.93)²] ≈ 3208.93
* **SS Between Proteins (SS Proteins):**
* SS Proteins = Σn(Protein Mean - Grand Mean)² (where n is the number of cell lines per protein, which is 3)
* 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
* **SS Error (SSE):**
* SSE = SST - SS Cells - SS Proteins
* SSE = 4165.73 - 3208.93 - 893.33 ≈ 63.47
**4. Calculate Degrees of Freedom (df)**
* df Cells = Number of cell lines - 1 = 3 - 1 = 2
* df Proteins = Number of proteins - 1 = 5 - 1 = 4
* df Error = (Number of cell lines - 1) * (Number of proteins - 1) = 2 * 4 = 8
* df Total = Total number of values - 1 = 15 - 1 = 14
**5. Calculate Mean Squares (MS)**
* MS Cells = SS Cells / df Cells = 3208.93 / 2 ≈ 1604.47
* MS Proteins = SS Proteins / df Proteins = 893.33 / 4 ≈ 223.33
* MS Error = SSE / df Error = 63.47 / 8 ≈ 7.93
**6. Calculate F-statistics**
* F Cells = MS Cells / MS Error = 1604.47 / 7.93 ≈ 202.33
* F Proteins = MS Proteins / MS Error = 223.33 / 7.93 ≈ 28.16
**7. Determine Critical F-values**
* **α = 0.05:**
* F_critical (2, 8) ≈ 4.46
* F_critical (4, 8) ≈ 3.84
* **α = 0.01:**
* F_critical (2, 8) ≈ 8.65
* F_critical (4, 8) ≈ 7.01
**8. Make Decisions**
**(a) Proteins**
* F Proteins (28.16) > F_critical (4, 8) at both 0.05 and 0.01.
* **Conclusion:** There is a significant difference in protein expression among the proteins.
**(b) Cell Lines**
* F Cells (202.33) > F_critical (2, 8) at both 0.05 and 0.01.
* **Conclusion:** There is a significant difference in protein expression among the cell lines.
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