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Aggregation Effects in Generalized Linear Models: a Biochemical Engineering Application

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Aggregation Effects in Generalized Linear Models: a Biochemical Engineering Application Iowa State University Capstones, Theses and Creative Components Dissertations Summer 2019 Aggregation Effects in Generalized Linear Models: A Biochemical Engineering Application Xiaojing Zhong Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/creativecomponents Part of the Engineering Commons Recommended Citation Zhong, Xiaojing, "Aggregation Effects in Generalized Linear Models: A Biochemical Engineering Application" (2019). Creative Components. 463. https://lib.dr.iastate.edu/creativecomponents/463 This Creative Component is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Creative Components by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Aggregation Effects in Generalized Linear Models: A Biochemical Engineering Application by Xiaojing Zhong Major: Statistics Program of Study Committee: Mark Kaiser, Major Professor Emily Berg Yumou Qiu Iowa State University Ames, Iowa 2019 ii TABLE OF CONTENTS Page TABLE OF CONTENTS .................................................................................................... ii LIST OF FIGURES ........................................................................................................... iii LIST OF TABLES .............................................................................................................. v CHAPTER 1. BACKGROUND ......................................................................................... 1 CHAPTER 2. EXPERIMENTAL SETUP ......................................................................... 2 2.1 TMS stimulator and coil .......................................................................................... 2 2.2 Experimental procedure ........................................................................................... 2 CHAPTER 3. TRADITIONAL ANALYSIS IN CHEMICAL ENGINEERING .............. 5 3.1 Standard analysis ..................................................................................................... 5 3.2 Problems with standard analysis .............................................................................. 7 CHAPTER 4. ANALYSIS USING GENERALIZED LINEAR MODEL ......................... 8 4.1 Generalized linear model ......................................................................................... 8 4.2 Examination of image effects ................................................................................ 16 4.3 Aggregate data in generalized linear model .......................................................... 19 CHAPTER 5. THEORETICAL ANALYSIS ON AGGREGATION EFFECTS ............ 22 5.1 Same α and same β for each time point ............................................................ 22 5.2 Same β but different α at each time point......................................................... 23 5.3 Different α and different β ................................................................................ 25 CHAPTER 6. EXAMINATION OF AGGREGATION EFFECTS ON DATA .............. 26 6.1 Visual examination ................................................................................................ 26 6.1.1 Visual examination on coefficients ............................................................... 26 6.1.2 Visual examination on responses .................................................................. 31 6.2 Likelihood ratio test ............................................................................................... 33 6.2.1 Same α and same β for each time point ......................................................... 33 6.2.2 Same β but different α for each time point .................................................... 34 6.3 Results of examination of aggregation effects on data .......................................... 35 CHAPTER 7. FUTURE WORK ...................................................................................... 36 REFERENCES ................................................................................................................. 37 iii LIST OF FIGURES Page Figure 2.1 Magstim 2002 TMS stimulator and figure-8 coil ........................................... 2 Figure 2.2 6-well plates used in the experiments where (1a) stimulated samples on plate 1; (1b) non-stimulated samples on plate 1; (2a) stimulated samples on plate 2; (2b) non-stimulated samples on plate 2 .......................... 3 Figure 2.3 Timeline of cell seeding and cell counting in the experiments ...................... 4 Figure 3.1 Normalized cell numbers in stimulated group ................................................ 6 Figure 3.2 Normalized cell numbers in non-stimulated group ........................................ 7 Figure 4.1 Scatterplot of the cell numbers for the stimulated samples with collagen substrate .......................................................................................................... 8 Figure 4.2 Scatterplot of the cell counts for the non-stimulated samples with collagen substrate ........................................................................................... 9 Figure 4.3 Log of group standard deviation against log of group mean for the stimulated samples with collagen substrate .................................................. 10 Figure 4.4 Log of group standard deviation against log of group mean for the non- stimulated samples with collagen substrate .................................................. 11 Figure 4.5 Log of cell count against time for the samples from stimulated group ........ 12 Figure 4.6 Log of cell count against time for the samples from non-stimulated group ............................................................................................................. 12 Figure 4.7 Expectation functions based on the samples from stimulated group ........... 14 Figure 4.8 Expectation functions based on the samples from non-stimulated group .... 14 Figure 4.9 Deviance residuals for the samples from stimulated group .......................... 15 Figure 4.10 Deviance residuals for the samples from non-stimulated group .................. 15 Figure 4.11 Expectation functions based on full data and subsets of data from stimulated group ........................................................................................... 16 Figure 4.12 Expectation functions based on full data and subsets of data from non- stimulated group ........................................................................................... 17 iv Figure 4.13 Expectation function and confidence interval for aggregate and unaggregated data from stimulated group .................................................... 20 Figure 4.14 Expectation function and confidence interval for aggregate and unaggregated data from non-stimulated group ............................................. 21 Figure 5.1 Expectation functions from individual images (“8 images” and “Average”) or the set of images (“Original”) for stimulated group ............. 32 Figure 5.2 Expectation functions from individual images (“8 images” and “Average”) or the set of images (“Original”) for non-stimulated group ...... 33 v LIST OF TABLES Page Table 3.1 Proliferation rate constant ( ) for the samples in stimulated and non- stimulated group. ............................................................................................ 5 α Table 4.1 Linear regression coefficients to find the relationship between the mean and variance .................................................................................................. 11 Table 4.2 Coefficients from generalized linear models for stimulated and non- stimulated groups with collagen substrate .................................................... 13 Table 4.3 The estimated based on full data and point Monte Carlo approximation to E and interval Monte Carlo approximation to E based on the subsetsγ0 of data .......................................................................... 18 γ0 γ0 Table 4.4 The estimated based on full data and point Monte Carlo approximation to E and interval Monte Carlo approximation to E based on the subsetsγ1 of data .......................................................................... 18 γ0 γ1 Table 4.5 The estimated based on full data and point Monte Carlo approximation to E and interval Monte Carlo approximation to E based on the subsetsϕ of data .......................................................................... 18 ϕ ϕ Table 4.6 Coefficients after averaging the cell counts from the samples in stimulated group) .......................................................................................... 19 Table 4.7 Coefficients after averaging the cell counts from the samples in non- stimulated group ........................................................................................... 20 Table 6.1 The coefficients from eight individual image regression for sample 1 in stimulated group ........................................................................................... 26 Table 6.2 The coefficients from eight individual image regression for sample 2 in stimulated group ........................................................................................... 27 Table 6.3 The coefficients from eight individual image regression for sample 3 in stimulated group ..........................................................................................
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