# Quantile Regression Approach for Analyzing Similarity of Gene Expressions under Multiple Biological Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Quantile Model for Gene Expression Data

- Random intercept model, $Z\left(t\right)=\left(1\right)$;
- Random slope model $Z\left(t\right)={(1,t)}^{\top}$.

#### 2.1. Estimation of Parameters

- 1.
- Obtain R bootstrap samples from the original data $\{{Y}_{i}\left({t}_{ij}\right),B\left({t}_{ij}\right),Z\left({t}_{ij}\right);j=1,2,\dots ,{k}_{i};$$i=1,2,\dots ,n\}$
- 2.
- Find the estimated values for the parameters ${\mathbf{\beta}}^{\left(\tau \right)},{\mathbf{\alpha}}^{\left(\tau \right)}$ and ${\mathbf{\sigma}}^{\left(\tau \right)}$ and then calculate the values of ${\widehat{\mathit{U}}}^{\left(\tau \right)}$ by using formula (11) from each bootstrap sample and denote the obtained values as ${\widehat{\mathbf{\beta}}}_{1}^{\left(\tau \right)},\dots ,{\widehat{\mathbf{\beta}}}_{R}^{\left(\tau \right)};$${\widehat{\mathbf{\alpha}}}_{1}^{\left(\tau \right)},\dots ,{\widehat{\mathbf{\alpha}}}_{R}^{\left(\tau \right)}$; ${\widehat{\mathbf{\sigma}}}_{1}^{\left(\tau \right)},\dots ,{\widehat{\mathbf{\sigma}}}_{R}^{\left(\tau \right)}$ and ${\widehat{\mathit{U}}}_{1}^{\left(\tau \right)},\dots ,{\widehat{\mathit{U}}}_{R}^{\left(\tau \right)}.$
- 3.
- Set ${\widehat{\mathbf{\varphi}}}_{r}^{\left(\tau \right)}={({\widehat{\mathbf{\beta}}}_{r}^{\left(\tau \right)\top},{\widehat{\mathit{U}}}_{r}^{\left(\tau \right)\top})}^{\top},{\widehat{\mathbf{\vartheta}}}_{r}^{\left(\tau \right)}={({\widehat{\mathbf{\beta}}}_{r}^{\left(\tau \right)\top},{\widehat{\mathbf{\alpha}}}_{r}^{\left(\tau \right)},{\widehat{\mathbf{\sigma}}}_{r}^{\left(\tau \right)})}^{\top},r=1,2\dots ,R$ and calculate the sample means of R bootstrap estimates for fixed effects parameters and random effects predictors ${\mathbf{\varphi}}^{\left(\tau \right)}={({\mathbf{\beta}}^{\left(\tau \right)\top},{\mathit{U}}^{\left(\tau \right)\top})}^{\top}$ and ${\mathbf{\vartheta}}_{r}^{\left(\tau \right)}={({\mathbf{\beta}}_{r}^{\left(\tau \right)\top},{\mathbf{\alpha}}_{r}^{\left(\tau \right)},{\mathbf{\sigma}}_{r}^{\left(\tau \right)})}^{\top}$$${\overline{\mathbf{\varphi}}}^{\left(\tau \right)}=\frac{1}{R}\sum _{r=1}^{R}{\widehat{\mathbf{\varphi}}}_{r}^{\left(\tau \right)},{\overline{\mathbf{\vartheta}}}^{\left(\tau \right)}=\frac{1}{R}\sum _{r=1}^{R}{\widehat{\mathbf{\vartheta}}}_{r}^{\left(\tau \right)}$$
- 4.
- Now, the bootstrap estimator for covariance matrix of estimators $({\widehat{\mathbf{\beta}}}^{\left(\tau \right)},{\widehat{\mathit{U}}}^{\left(\tau \right)})$ can be written as$$\begin{array}{c}\hfill {\widehat{\mathit{V}}}_{{\mathbf{\varphi}}^{\left(\tau \right)}}=\widehat{\mathrm{cov}}({\widehat{\mathbf{\beta}}}^{\left(\tau \right)},{\widehat{\mathit{U}}}^{\left(\tau \right)})=\frac{1}{R-1}\sum _{r=1}^{R}({\widehat{\mathbf{\varphi}}}_{r}^{\left(\tau \right)}-{\overline{\mathbf{\varphi}}}^{\left(\tau \right)}){({\widehat{\mathbf{\varphi}}}_{r}^{\left(\tau \right)}-{\overline{\mathbf{\varphi}}}^{\left(\tau \right)})}^{\top},\end{array}$$$$\begin{array}{c}\hfill {\widehat{\mathit{V}}}_{{\mathbf{\vartheta}}^{\left(\tau \right)}}=\widehat{\mathrm{cov}}({\mathbf{\beta}}_{r}^{\left(\tau \right)\top},{\mathbf{\alpha}}_{r}^{\left(\tau \right)},{\mathbf{\sigma}}_{r}^{\left(\tau \right)})=\frac{1}{R-1}\sum _{r=1}^{R}({\widehat{\mathbf{\vartheta}}}_{r}^{\left(\tau \right)}-{\overline{\mathbf{\vartheta}}}^{\left(\tau \right)}){({\widehat{\mathbf{\vartheta}}}_{r}^{\left(\tau \right)}-{\overline{\mathbf{\vartheta}}}^{\left(\tau \right)})}^{\top}\end{array}$$

**lqmm**’ package developed by Geraci and Bottai [17] in the

**R**programming environment. Generally, this ‘

**lqmm**’ package is used to estimate conditional quantile functions with random effects in linear quantile mixed models.

#### 2.2. Test of the Similarity of Quantile Functions for Two Gene Expressions

## 3. Simulation

- Thirty-five equally spaced timepoints between [0, 1] are considered for 50 samples. The data were generated using the following mixed model. The true model is assumed as follows:$${Y}_{i}\left(t\right)={f}_{0}\left(t\right)+{U}_{i}\left(t\right)+{\u03f5}_{i}\left(t\right);\hspace{0.17em}\hspace{0.17em}t\in [0,1],$$
- Random effects components are assumed to follow a normal distribution with a mean of zero and standard deviation of two, i.e., $({U}_{i}\sim N(0,4))$.
- The following i.i.d errors are considered for true models Equation (20).
- 1.
- Model 1: ${\u03f5}_{i}\left(t\right)\sim $ Laplace(0,1)
- 2.
- Model 2: ${\u03f5}_{i}\left(t\right)\sim $ Normal(0,1)
- 3.
- Model 3: ${\u03f5}_{i}\left(t\right)\sim $ Skew Normal(0,1,1)
- 4.
- Model 4: ${\u03f5}_{i}\left(t\right)\sim $ Skew t(0,1,1,4)
- 5.
- Model 5: ${\u03f5}_{i}\left(t\right)\sim $ t(3)

#### 3.1. Model and Parameter Estimation

- Generate data for $n=50$ samples using Equation (20).
- Basis functions for Model (21) are considered as:
- 1.
- Random intercept model: $B\left(t\right)={(1,t,{t}^{2},{t}^{3})}^{\top}$, $Z\left(t\right)=\left(1\right)$.
- 2.
- Random slope model: $B\left(t\right)={(1,t,{t}^{2},{t}^{3})}^{\top}$, $Z\left(t\right)=(1,t)$.

- Forthe random slope model, we consider normal random effects with a diagonal variance–covariance matrix. A Gauss–Hermite quadrature with seven noves is considered to approximate the marginal log-likelihood of Equation (10).
- All parameters are estimated at median ($\tau =0.50$).
- A total of 500 bootstrap replications are considered for the estimation of the covariance matrix of the estimators $({\widehat{\mathbf{\beta}}}^{\left(\tau \right)},{\widehat{\mathbf{U}}}^{\left(\tau \right)})$.

#### 3.2. Simulation Results for Parameter Estimation

#### 3.2.1. Parameter Estimates of Random Intercept QR Model

- Estimates of parameters in Model 1: ${\widehat{\mathbf{\beta}}}^{\left(\tau \right)}={[1.19,-5.06,68.86,-53.29]}^{\top}$, ${\widehat{\mathbf{\alpha}}}^{\left(\tau \right)}=1.23$, ${\widehat{\sigma}}^{\left(\tau \right)}=0.56$, ${\widehat{\mathsf{\Psi}}}^{\left(\tau \right)}=1.52$.
- Estimates of parameters in Model 2: ${\widehat{\mathbf{\beta}}}^{\left(\tau \right)}={[1.68,-7.42,74.74,-56.87]}^{\top}$, ${\widehat{\mathbf{\alpha}}}^{\left(\tau \right)}=1.01$, ${\widehat{\sigma}}^{\left(\tau \right)}=0.43$, ${\widehat{\mathsf{\Psi}}}^{\left(\tau \right)}=1.02$.
- Estimates of parameters in Model 3: ${\widehat{\mathbf{\beta}}}^{\left(\tau \right)}={[2.03,-7.46,74.81,-57.17]}^{\top}$, ${\widehat{\mathbf{\alpha}}}^{\left(\tau \right)}=1.14$, ${\widehat{\sigma}}^{\left(\tau \right)}=0.36$, ${\widehat{\mathsf{\Psi}}}^{\left(\tau \right)}=1.29$.
- Estimates of parameters in Model 4: ${\widehat{\mathbf{\beta}}}^{\left(\tau \right)}={[1.88,-7.85,76.28,-57.92]}^{\top}$, ${\widehat{\mathbf{\alpha}}}^{\left(\tau \right)}=1.41$, ${\widehat{\sigma}}^{\left(\tau \right)}=0.49$, ${\widehat{\mathsf{\Psi}}}^{\left(\tau \right)}=1.98$.
- Estimates of parameters in Model 5: ${\widehat{\mathbf{\beta}}}^{\left(\tau \right)}={[1.29,-8.70,76.78,-57.78]}^{\top}$, ${\widehat{\mathbf{\alpha}}}^{\left(\tau \right)}=1.42$, ${\widehat{\sigma}}^{\left(\tau \right)}=0.57$, ${\widehat{\mathsf{\Psi}}}^{\left(\tau \right)}=2.01$.

#### 3.2.2. Parameter Estimates of Random Slope QR Model

- Estimates of parameters in Model 1: ${\widehat{\mathbf{\beta}}}^{\left(\tau \right)}={[1.27,-5.08,68.86,-53.31]}^{\top}$, ${\widehat{\mathbf{\alpha}}}^{\left(\tau \right)}={[1.35,0.67]}^{\top}$, ${\widehat{\sigma}}^{\left(\tau \right)}=0.54$, ${\widehat{\mathsf{\Psi}}}_{\mathrm{intercept}}^{\left(\tau \right)}=1.83,{\widehat{\mathsf{\Psi}}}_{\mathrm{time}}^{\left(\tau \right)}=0.45$.
- Estimates of parameters in Model 2: ${\widehat{\mathbf{\beta}}}^{\left(\tau \right)}={[0.98,-6.43,72.80,-55.64]}^{\top}$, ${\widehat{\mathbf{\alpha}}}^{\left(\tau \right)}={[1.29,0.68]}^{\top}$, ${\widehat{\sigma}}^{\left(\tau \right)}=0.41$, ${\widehat{\mathsf{\Psi}}}_{\mathrm{intercept}}^{\left(\tau \right)}=1.68,{\widehat{\mathsf{\Psi}}}_{\mathrm{time}}^{\left(\tau \right)}=0.46$.
- Estimates of parameters in Model 3: ${\widehat{\mathbf{\beta}}}^{\left(\tau \right)}={[1.92,-7.43,74.85,-57.15]}^{\top}$, ${\widehat{\mathbf{\alpha}}}^{\left(\tau \right)}={[1.79,0.73]}^{\top}$, ${\widehat{\sigma}}^{\left(\tau \right)}=0.35$, ${\widehat{\mathsf{\Psi}}}_{\mathrm{intercept}}^{\left(\tau \right)}=3.20,{\widehat{\mathsf{\Psi}}}_{\mathrm{time}}^{\left(\tau \right)}=0.53$.
- Estimates of parameters in Model 4: ${\widehat{\mathbf{\beta}}}^{\left(\tau \right)}={[2.30,-7.83,76.31,-57.87]}^{\top}$, ${\widehat{\mathbf{\alpha}}}^{\left(\tau \right)}={[1.67,1.06]}^{\top}$, ${\widehat{\sigma}}^{\left(\tau \right)}=0.46$, ${\widehat{\mathsf{\Psi}}}_{\mathrm{intercept}}^{\left(\tau \right)}=2.79,{\widehat{\mathsf{\Psi}}}_{\mathrm{time}}^{\left(\tau \right)}=1.13$.

- Estimates of parameters in Model 5: ${\widehat{\mathbf{\beta}}}^{\left(\tau \right)}={[1.25,-8.53,76.79,-57.92]}^{\top}$, ${\widehat{\mathbf{\alpha}}}^{\left(\tau \right)}={[1.47,0.53]}^{\top}$, ${\widehat{\sigma}}^{\left(\tau \right)}=0.57$, ${\widehat{\mathsf{\Psi}}}_{\mathrm{intercept}}^{\left(\tau \right)}=2.16,{\widehat{\mathsf{\Psi}}}_{\mathrm{time}}^{2}=0.29$.

#### 3.3. Power Analysis of Proposed Test Statistic

## 4. Application: Gene Expression Data

#### 4.1. Data

#### 4.2. Exploratory Analysis

#### 4.3. Model and Parameter Estimates

- Random Intercept Model: $B\left(t\right)={(1,t,{t}^{2},{t}^{3},{t}^{4})}^{\top}$ and $Z\left(t\right)=\left(1\right)$.
- Random Slope Model: $B\left(t\right)={(1,t,{t}^{2},{t}^{3},{t}^{4})}^{\top}$ and $Z\left(t\right)={(1,t)}^{\top}$.

#### 4.4. Test of Gene Similarity

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Estimated median function and C.I. (

**left**); Estimated median and sample median (

**right**) for Model 1.

**Figure 2.**Estimated median function and C.I. (

**left**); Estimated median and sample median (

**right**) for Model 2.

**Figure 3.**Estimated median function and C.I. (

**left**); Estimated median and sample median (

**right**) for Model 3.

**Figure 4.**Estimated median function and C.I. (

**left**); Estimated median and sample median (

**right**) for Model 4.

**Figure 5.**Estimated median function and C.I. (

**left**); Estimated median and sample median (

**right**) for Model 5.

**Figure 6.**Estimated median function and C.I. (

**left**); Estimated median and sample median (

**right**) for Model 1.

**Figure 7.**Estimated median function and C.I. (

**left**); Estimated median and sample median (

**right**) for Model 2.

**Figure 8.**Estimated median function and C.I. (

**left**); Estimated median and sample median (

**right**) for Model 3.

**Figure 9.**Estimated median function and C.I. (

**left**); Estimated median and sample median (

**right**) for Model 4.

**Figure 10.**Estimated median function and C.I. (

**left**); Estimated median and sample median (

**right**) for Model 5.

**Figure 13.**Box-plots of gene expressions PA2975(rluc)(B3) regarding time (

**left**) and conditions (

**right**).

**Figure 17.**Quantile functions and sample quantile functions of PA2975(rluc)(B3): lower quartile (

**left**); median (

**center**); upper quartile (

**right**).

**Figure 18.**Quantile functions and sample quantile functions of PA0573(E6): lower quartile (

**left**); median (

**center**); upper quartile (

**right**).

**Table 1.**Empirical Powers of ${\mathbf{\chi}}_{hs}^{2\left(\tau \right)}$ for intercept models and slope models with $\tau =0.25$.

(a) Intercept Models | (b) Slope Models | ||||||
---|---|---|---|---|---|---|---|

Models | Distance | $\mathit{n}=30$ | $\mathit{n}=50$ | $\mathit{n}=75$ | $\mathit{n}=30$ | $\mathit{n}=50$ | $\mathit{n}=75$ |

0.00 | 0.055 | 0.052 | 0.051 | 0.042 | 0.049 | 0.052 | |

0.50 | 0.077 | 0.083 | 0.073 | 0.066 | 0.066 | 0.054 | |

1.00 | 0.144 | 0.175 | 0.174 | 0.122 | 0.151 | 0.197 | |

Model 1 | 1.50 | 0.293 | 0.347 | 0.397 | 0.234 | 0.320 | 0.408 |

2.00 | 0.476 | 0.576 | 0.628 | 0.420 | 0.551 | 0.659 | |

2.50 | 0.693 | 0.777 | 0.828 | 0.639 | 0.752 | 0.860 | |

3.00 | 0.837 | 0.900 | 0.954 | 0.808 | 0.873 | 0.943 | |

0.00 | 0.055 | 0.047 | 0.049 | 0.044 | 0.051 | 0.047 | |

0.50 | 0.094 | 0.065 | 0.077 | 0.074 | 0.063 | 0.078 | |

1.00 | 0.163 | 0.171 | 0.215 | 0.122 | 0.155 | 0.208 | |

Model 2 | 1.50 | 0.325 | 0.365 | 0.453 | 0.242 | 0.315 | 0.443 |

2.00 | 0.533 | 0.597 | 0.709 | 0.430 | 0.556 | 0.686 | |

2.50 | 0.728 | 0.811 | 0.868 | 0.653 | 0.774 | 0.851 | |

3.00 | 0.871 | 0.922 | 0.958 | 0.826 | 0.899 | 0.944 | |

0.00 | 0.068 | 0.057 | 0.051 | 0.059 | 0.050 | 0.046 | |

0.50 | 0.108 | 0.082 | 0.086 | 0.078 | 0.065 | 0.080 | |

1.00 | 0.194 | 0.204 | 0.218 | 0.138 | 0.153 | 0.178 | |

Model 3 | 1.50 | 0.349 | 0.417 | 0.494 | 0.255 | 0.346 | 0.403 |

2.00 | 0.563 | 0.668 | 0.739 | 0.449 | 0.601 | 0.661 | |

2.50 | 0.756 | 0.847 | 0.897 | 0.653 | 0.797 | 0.842 | |

3.00 | 0.885 | 0.957 | 0.964 | 0.831 | 0.920 | 0.950 | |

0.00 | 0.066 | 0.058 | 0.053 | 0.057 | 0.046 | 0.054 | |

0.50 | 0.086 | 0.084 | 0.086 | 0.072 | 0.057 | 0.061 | |

1.00 | 0.159 | 0.172 | 0.194 | 0.133 | 0.148 | 0.173 | |

Model 4 | 1.50 | 0.298 | 0.375 | 0.428 | 0.252 | 0.326 | 0.389 |

2.00 | 0.523 | 0.621 | 0.697 | 0.423 | 0.572 | 0.662 | |

2.50 | 0.717 | 0.816 | 0.865 | 0.669 | 0.774 | 0.842 | |

3.00 | 0.871 | 0.933 | 0.952 | 0.818 | 0.901 | 0.942 | |

0.00 | 0.056 | 0.053 | 0.049 | 0.057 | 0.043 | 0.046 | |

0.50 | 0.085 | 0.082 | 0.085 | 0.085 | 0.062 | 0.071 | |

1.00 | 0.144 | 0.167 | 0.187 | 0.149 | 0.157 | 0.195 | |

Model 5 | 1.50 | 0.270 | 0.336 | 0.402 | 0.253 | 0.323 | 0.422 |

2.00 | 0.460 | 0.558 | 0.671 | 0.448 | 0.553 | 0.675 | |

2.50 | 0.651 | 0.762 | 0.844 | 0.634 | 0.748 | 0.851 | |

3.00 | 0.806 | 0.890 | 0.947 | 0.786 | 0.889 | 0.949 |

**Table 2.**Empirical Powers of ${\mathbf{\chi}}_{hs}^{2\left(\tau \right)}$ for intercept models and slope models with $\tau =0.50$.

(a) Intercept Models | (b) Slope Models | ||||||
---|---|---|---|---|---|---|---|

Models | Distance | $\mathit{n}=30$ | $\mathit{n}=50$ | $\mathit{n}=75$ | $\mathit{n}=30$ | $\mathit{n}=50$ | $\mathit{n}=75$ |

0.00 | 0.071 | 0.047 | 0.051 | 0.051 | 0.057 | 0.053 | |

0.50 | 0.095 | 0.091 | 0.086 | 0.076 | 0.087 | 0.072 | |

1.00 | 0.193 | 0.240 | 0.270 | 0.154 | 0.182 | 0.227 | |

Model 1 | 1.50 | 0.360 | 0.489 | 0.574 | 0.313 | 0.413 | 0.505 |

2.00 | 0.597 | 0.751 | 0.812 | 0.521 | 0.664 | 0.774 | |

2.50 | 0.807 | 0.903 | 0.940 | 0.720 | 0.849 | 0.913 | |

3.00 | 0.919 | 0.976 | 0.991 | 0.874 | 0.951 | 0.981 | |

0.00 | 0.066 | 0.049 | 0.052 | 0.048 | 0.050 | 0.046 | |

0.50 | 0.092 | 0.089 | 0.100 | 0.080 | 0.069 | 0.059 | |

1.00 | 0.190 | 0.239 | 0.318 | 0.158 | 0.176 | 0.227 | |

Model 2 | 1.50 | 0.396 | 0.479 | 0.635 | 0.317 | 0.397 | 0.491 |

2.00 | 0.638 | 0.730 | 0.854 | 0.536 | 0.664 | 0.761 | |

2.50 | 0.849 | 0.922 | 0.967 | 0.749 | 0.857 | 0.927 | |

3.00 | 0.944 | 0.977 | 0.990 | 0.876 | 0.952 | 0.978 | |

0.00 | 0.078 | 0.047 | 0.048 | 0.065 | 0.051 | 0.057 | |

0.50 | 0.102 | 0.105 | 0.096 | 0.087 | 0.063 | 0.073 | |

1.00 | 0.211 | 0.240 | 0.320 | 0.152 | 0.196 | 0.226 | |

Model 3 | 1.50 | 0.383 | 0.511 | 0.615 | 0.310 | 0.414 | 0.488 |

2.00 | 0.628 | 0.770 | 0.842 | 0.493 | 0.665 | 0.752 | |

2.50 | 0.820 | 0.924 | 0.955 | 0.711 | 0.850 | 0.911 | |

3.00 | 0.937 | 0.990 | 0.992 | 0.872 | 0.954 | 0.976 | |

0.00 | 0.076 | 0.057 | 0.051 | 0.044 | 0.045 | 0.047 | |

0.50 | 0.101 | 0.080 | 0.102 | 0.084 | 0.070 | 0.071 | |

1.00 | 0.202 | 0.223 | 0.291 | 0.151 | 0.191 | 0.226 | |

Model 4 | 1.50 | 0.379 | 0.510 | 0.606 | 0.328 | 0.436 | 0.508 |

2.00 | 0.626 | 0.754 | 0.847 | 0.559 | 0.679 | 0.767 | |

2.50 | 0.835 | 0.911 | 0.952 | 0.755 | 0.855 | 0.909 | |

3.00 | 0.934 | 0.977 | 0.985 | 0.881 | 0.951 | 0.975 | |

0.00 | 0.058 | 0.055 | 0.052 | 0.048 | 0.056 | 0.048 | |

0.50 | 0.092 | 0.084 | 0.089 | 0.079 | 0.074 | 0.080 | |

1.00 | 0.183 | 0.222 | 0.273 | 0.157 | 0.166 | 0.246 | |

Model 5 | 1.50 | 0.374 | 0.491 | 0.587 | 0.316 | 0.420 | 0.513 |

2.00 | 0.608 | 0.724 | 0.836 | 0.522 | 0.672 | 0.786 | |

2.50 | 0.796 | 0.897 | 0.942 | 0.741 | 0.851 | 0.924 | |

3.00 | 0.922 | 0.966 | 0.983 | 0.874 | 0.936 | 0.982 |

**Table 3.**Empirical Powers of ${\mathbf{\chi}}_{hs}^{2\left(\tau \right)}$ for intercept models and slope models with $\tau =0.75$.

(a) Intercept Models | (b) Slope Models | ||||||
---|---|---|---|---|---|---|---|

Models | Distance | $\mathit{n}=30$ | $\mathit{n}=50$ | $\mathit{n}=75$ | $\mathit{n}=30$ | $\mathit{n}=50$ | $\mathit{n}=75$ |

0.00 | 0.073 | 0.056 | 0.047 | 0.069 | 0.058 | 0.044 | |

0.50 | 0.098 | 0.078 | 0.069 | 0.081 | 0.071 | 0.071 | |

1.00 | 0.163 | 0.170 | 0.197 | 0.150 | 0.158 | 0.177 | |

Model 1 | 1.50 | 0.303 | 0.359 | 0.415 | 0.285 | 0.320 | 0.396 |

2.00 | 0.499 | 0.572 | 0.661 | 0.447 | 0.551 | 0.666 | |

2.50 | 0.686 | 0.777 | 0.852 | 0.639 | 0.758 | 0.856 | |

3.00 | 0.818 | 0.903 | 0.955 | 0.795 | 0.894 | 0.943 | |

0.00 | 0.058 | 0.050 | 0.050 | 0.050 | 0.057 | 0.048 | |

0.50 | 0.083 | 0.066 | 0.082 | 0.064 | 0.070 | 0.074 | |

1.00 | 0.164 | 0.170 | 0.198 | 0.139 | 0.156 | 0.199 | |

Model 2 | 1.50 | 0.332 | 0.380 | 0.455 | 0.251 | 0.346 | 0.442 |

2.00 | 0.535 | 0.634 | 0.724 | 0.449 | 0.567 | 0.696 | |

2.50 | 0.740 | 0.816 | 0.881 | 0.655 | 0.771 | 0.863 | |

3.00 | 0.880 | 0.923 | 0.960 | 0.809 | 0.899 | 0.941 | |

0.00 | 0.075 | 0.046 | 0.054 | 0.077 | 0.046 | 0.050 | |

0.50 | 0.101 | 0.086 | 0.084 | 0.098 | 0.078 | 0.081 | |

1.00 | 0.173 | 0.202 | 0.226 | 0.178 | 0.169 | 0.214 | |

Model 3 | 1.50 | 0.347 | 0.440 | 0.486 | 0.303 | 0.371 | 0.445 |

2.00 | 0.539 | 0.677 | 0.757 | 0.463 | 0.609 | 0.690 | |

2.50 | 0.739 | 0.864 | 0.910 | 0.663 | 0.817 | 0.855 | |

3.00 | 0.873 | 0.940 | 0.958 | 0.817 | 0.921 | 0.951 | |

0.00 | 0.066 | 0.050 | 0.051 | 0.055 | 0.044 | 0.044 | |

0.50 | 0.079 | 0.081 | 0.082 | 0.077 | 0.055 | 0.072 | |

1.00 | 0.166 | 0.188 | 0.203 | 0.132 | 0.163 | 0.205 | |

Model 4 | 1.50 | 0.319 | 0.388 | 0.451 | 0.273 | 0.360 | 0.446 |

2.00 | 0.523 | 0.638 | 0.680 | 0.468 | 0.610 | 0.726 | |

2.50 | 0.717 | 0.819 | 0.873 | 0.679 | 0.803 | 0.887 | |

3.00 | 0.868 | 0.946 | 0.957 | 0.831 | 0.925 | 0.966 | |

0.00 | 0.046 | 0.046 | 0.054 | 0.056 | 0.051 | 0.047 | |

0.50 | 0.070 | 0.066 | 0.081 | 0.070 | 0.073 | 0.076 | |

1.00 | 0.131 | 0.154 | 0.187 | 0.136 | 0.143 | 0.177 | |

Model 5 | 1.50 | 0.264 | 0.333 | 0.400 | 0.281 | 0.320 | 0.413 |

2.00 | 0.456 | 0.567 | 0.644 | 0.449 | 0.552 | 0.688 | |

2.50 | 0.664 | 0.773 | 0.831 | 0.633 | 0.757 | 0.867 | |

3.00 | 0.833 | 0.913 | 0.940 | 0.817 | 0.881 | 0.948 |

Code | Name | Protein | Ratio | Remarks |
---|---|---|---|---|

A6 | PA5283 | Probable Trancriptional regulator | 99.68% | 48% similar to putative transcriptional regulator (Bacilus subtilis) |

B3 | PA2975 (rluc) | Ribosomal large subunit | 99.68% | Transcription, RNA processing and degradation |

B4 | PA4991 | Hypothetical Protein | 100% | Unknown |

B5 | PA5237 | Conserved hypothetical Protein | 100% | 87% similar to hypothetical yigC gene product of E. coli |

C4 | PA0287 (gpuP) | 3-guanidinopropionate transport protein | 100% | Transport of small molecules |

D1 | PA3115 (fimV) | Motility protein FimV | 100% | Membrane proteins; Motility and Attachment |

D2 | PA3879 (narL) | Two-component response regulator NarL | 99.67% | 74% similar to E. coli NarL protein |

D3 | PA0894 | Hypothetical Protein | 99.02% | Unknown |

E5 | PA1875 | Probable outer membrane protein precursor | 100% | 41% similar to alkaline protease secretion protein AprF |

E6 | PA0573 | Hypothetical Protein | 100% | Unknown |

F2 | PA3902 | Hypothetical Protein | 100% | Unknown |

F3 | PA3212 | Probable ATP-binding component of ABC transporter | 100% | 65% similar to putative amino acid abc transporter, ATP-binding Protein (Helicobacter Pylori J99) |

F5 | PA2997 (nqrC) | Na+translocating NADH: ubiquinone oxidoreductase subunit Nrq3 | 100% | Energy metabolism |

G2 | PA0649 (trpG) | Anthranilate synthase component II | 100% | Energy metabolism; Biosynthesis of co-factors, prosthetic groups and carriers; Amino acid biosynthesis and metabolism |

G5 | PA1748 | Probable enoyl-CoA hydratase/isomerase | 98.20% | 61% similar to putative enoyl-coA hydratase EchA3 (Mycobacterium tuberculosis) |

G6 | PA3771 | Probable trancriptional regulator | 99.22% | 54% similar to a region of putative regulatory protein (Streptomyces coelicolor) |

H3 | PA1841 | Hypothetical protein | 100% | 43% similar to hypothetical yeaK gene product of (E. coli) |

S70 | $\sigma $70 | $\sigma $ factor | As a control |

PA2975(rluc)(B3) | PA0573(E6) | |||||
---|---|---|---|---|---|---|

Estimate | $\mathbf{\tau}=\mathbf{0.25}$ | $\mathbf{\tau}=\mathbf{0.50}$ | $\mathbf{\tau}=\mathbf{0.75}$ | $\mathbf{\tau}=\mathbf{0.25}$ | $\mathbf{\tau}=\mathbf{0.50}$ | $\mathbf{\tau}=\mathbf{0.75}$ |

${\beta}_{0}$ | 2.7592 | 2.843 | 2.925 | 2.569158 | 2.625809 | 2.785331 |

${\beta}_{1}$ | 0.54199 | 0.53962 | 0.53468 | 0.465631 | 0.494351 | 0.4591522 |

${\beta}_{2}$ | −0.054218 | −0.054119 | -0.052799 | −0.039942 | −0.042743 | −0.038572 |

${\beta}_{3}$ | 0.002192 | 0.0021183 | 0.0020676 | 0.001215 | 0.001262 | 0.001171 |

${\beta}_{4}$ | −0.000031 | −0.000029 | −0.000029 | −0.000011 | −0.000008 | −0.000009 |

$\widehat{\sigma}$ | 0.068 | 0.096 | 0.072 | 0.084 | 0.101 | 0.076 |

AIC | 43.43 | 143.56 | 166.91 | 466.15 | 241.57 | 252.46 |

Gene | A6 | B3 | B4 | B5 | C4 | D1 | D2 | D3 | E5 | E6 | F2 | F3 | F5 | G2 | G5 | G6 | H3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

B3 | 222.58 | ||||||||||||||||

B4 | 145.73 | 17.46 | |||||||||||||||

B5 | 331.40 | 53.15 | 62.8 | ||||||||||||||

C4 | 41.58 | 21.14 | 20.96 | 54.61 | |||||||||||||

D1 | 29.88 | 127.88 | 64.77 | 176.25 | 66.30 | ||||||||||||

D2 | 5.78 | 77.32 | 79.50 | 105.09 | 29.10 | 29.56 | |||||||||||

D3 | 12.93 | 62.89 | 68.93 | 145.2 | 22.02 | 29.05 | 22.91 | ||||||||||

E5 | 18.15 | 35.27 | 41.42 | 49.32 | 27.38 | 18.92 | 24.82 | 26.35 | |||||||||

E6 | 159.01 | 39.13 | 34.87 | 190.01 | 14.69 | 78.55 | 51.99 | 28.54 | 20.33 | ||||||||

F2 | 35.29 | 33.85 | 34.91 | 54.04 | 29.13 | 33.34 | 49.49 | 15.86 | 49.9 | 17.52 | |||||||

F3 | 9.90 | 43.38 | 66.94 | 92.28 | 11.63 | 25.66 | 18.09 | 11.51 | 11.01 | 24.65 | 36.72 | ||||||

F5 | 120.18 | 39.42 | 19.50 | 37.26 | 31.58 | 71.71 | 93.65 | 88.4 | 52.68 | 55.69 | 43.79 | 74.78 | |||||

G2 | 185.55 | 15.76 | 53.09 | 54.37 | 11.47 | 105.15 | 53.06 | 39.77 | 27.35 | 31.29 | 19.38 | 25.87 | 51.5 | ||||

G5 | 104.23 | 25.73 | 5.68 | 62.91 | 11.67 | 71.79 | 61.42 | 40.07 | 28.85 | 11.2 | 36.9 | 39.48 | 25.01 | 52.04 | |||

G6 | 62.10 | 10.25 | 28.59 | 36.92 | 8.99 | 82.18 | 35.43 | 35.28 | 16.29 | 13.19 | 33.96 | 19.79 | 41.33 | 5.85 | 22.08 | ||

H3 | 104.05 | 8.72 | 7.82 | 35.07 | 21.29 | 78.07 | 71.37 | 45.62 | 36.34 | 9.94 | 25.01 | 39.33 | 24.02 | 25.64 | 6.22 | 17.28 | |

$\sigma $70 | 55.17 | 59.36 | 29.27 | 75.08 | 55.16 | 20.55 | 41.38 | 34.36 | 23.2 | 51.65 | 56.12 | 39.25 | 51.18 | 82.15 | 28.1 | 59.18 | 40.18 |

Gene | A6 | A6 | A6 | B3 | B3 | B4 | B4 | C4 | C4 |
---|---|---|---|---|---|---|---|---|---|

Gene | D2 | D3 | F3 | G6 | H3 | G5 | H3 | F3 | G2 |

Distance | 5.78 | 12.93 | 9.90 | 10.25 | 8.72 | 5.68 | 7.82 | 11.63 | 11.47 |

Gene | C4 | C4 | D3 | E5 | E6 | E6 | E6 | G2 | G5 |

Gene | G5 | G6 | F3 | F3 | G5 | G6 | H3 | G6 | H3 |

Distance | 11.67 | 8.99 | 11.51 | 11.01 | 11.20 | 13.19 | 9.94 | 5.85 | 6.22 |

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## Share and Cite

**MDPI and ACS Style**

Deng, D.; Chowdhury, M.H.
Quantile Regression Approach for Analyzing Similarity of Gene Expressions under Multiple Biological Conditions. *Stats* **2022**, *5*, 583-605.
https://doi.org/10.3390/stats5030036

**AMA Style**

Deng D, Chowdhury MH.
Quantile Regression Approach for Analyzing Similarity of Gene Expressions under Multiple Biological Conditions. *Stats*. 2022; 5(3):583-605.
https://doi.org/10.3390/stats5030036

**Chicago/Turabian Style**

Deng, Dianliang, and Mashfiqul Huq Chowdhury.
2022. "Quantile Regression Approach for Analyzing Similarity of Gene Expressions under Multiple Biological Conditions" *Stats* 5, no. 3: 583-605.
https://doi.org/10.3390/stats5030036