Multivariate regression



In [1]:

    
%pylab inline









    



Populating the interactive namespace from numpy and matplotlib



In [2]:

    
import pandas as pd



In [3]:

    
mypath = 'Cell_types.xlsx'
xls = pd.read_excel(mypath)



In [4]:

    
xls.head()









    Out[4]:






  
    
      
      CellID
      Vrest
      InputR
      mbTau
      SagRatio
      MaxAPfreq
      Animal Weight (g)
      Animal Age
      Gender
      AP peak (mv)
      AP_thr (mV)
    
  
  
    
      0
      16.08.30.0.5
      -64.8039
      123.611
      21.766200
      1.31594
      5
      23.1
      8 weeks
      male
      NaN
      NaN
    
    
      1
      16.08.31.0.6
      -74.9081
      152.837
      23.242900
      1.17876
      17
      24.6
      8 weeks
      male
      NaN
      NaN
    
    
      2
      16.08.31.1.2
      -74.4443
      221.895
      27.983400
      1.17550
      7
      24.6
      8 weeks
      male
      NaN
      NaN
    
    
      3
      16.09.01.0.1
      -75.8549
      294.484
      31.245583
      1.10705
      29
      26.0
      8 weeks
      female
      NaN
      NaN
    
    
      4
      16.09.01.0.2
      -66.3612
      271.674
      34.679290
      1.06450
      25
      26.0
      8 weeks
      female
      NaN
      NaN



In [5]:

    
xls.InputR









    Out[5]:





0    123.6110
1    152.8370
2    221.8950
3    294.4840
4    271.6740
5    137.2340
6     69.2104
7     97.5294
Name: InputR, dtype: float64



In [6]:

    
xls['Vrest'].mean()









    Out[6]:





-69.163162499999999



In [7]:

    
xls['Vrest'].unique() # get NumPy array









    Out[7]:





array([-64.8039, -74.9081, -74.4443, -75.8549, -66.3612, -68.7208,
       -66.2903, -61.9218])

Let's evaluate how much the membrane potential depends on Input resistance and membrane time constant and the sag ratio. We will create the following multivariate function:

$f(k;x) = k_0 + k_1x_1 + k_2x_2 + k_3x_3$

where k is a vector or parameters (contants) and x is a vector of independent variables (i.e x_1 is the input resistance x_2 is membrane time constant and x_3 the sag ratio)



In [8]:

    
x = xls[['InputR', 'SagRatio','mbTau']]
y = xls[['Vrest']]



In [9]:

    
# import standard regression models (sm)
import statsmodels.api as sm



In [10]:

    
X1 = sm.add_constant(x) # k0, k1, k2 and k3...



In [11]:

    
# get estimation
est = sm.OLS(y, X1).fit() # ordinary least square regression



In [12]:

    
est.summary()









    



/usr/lib/python2.7/dist-packages/scipy/stats/stats.py:1293: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=8
  int(n))






    Out[12]:





OLS Regression Results

  Dep. Variable:           Vrest         R-squared:             0.662


  Model:                    OLS          Adj. R-squared:        0.408


  Method:              Least Squares     F-statistic:           2.611


  Date:              Tue, 06 Sep 2016    Prob (F-statistic):    0.188 


  Time:                  22:39:27        Log-Likelihood:      -19.755


  No. Observations:            8         AIC:                   47.51


  Df Residuals:                4         BIC:                   47.83


  Df Model:                    3                                     




              coef      std err       t       P>|t|  [95.0% Conf. Int.] 


  const       -65.9312     21.957     -3.003   0.040   -126.894    -4.969


  InputR       -0.1582      0.069     -2.279   0.085     -0.351     0.034


  SagRatio    -14.6860     16.502     -0.890   0.424    -60.503    31.131


  mbTau         1.6515      0.924      1.788   0.148     -0.913     4.217




  Omnibus:         2.275    Durbin-Watson:         2.142


  Prob(Omnibus):   0.321    Jarque-Bera (JB):      0.762


  Skew:           -0.752    Prob(JB):              0.683


  Kurtosis:        2.835    Cond. No.           3.50e+03

r_squared is not very large... but coef gives us the values of K0, K1, k2, and K3 to plug into the equation. Based on the std err InputResistance and mbTau are more important than SagRatio to determine the AP.



In [ ]:

	CellID	Vrest	InputR	mbTau	SagRatio	MaxAPfreq	Animal Weight (g)	Animal Age	Gender	AP peak (mv)	AP_thr (mV)
0	16.08.30.0.5	-64.8039	123.611	21.766200	1.31594	5	23.1	8 weeks	male	NaN	NaN
1	16.08.31.0.6	-74.9081	152.837	23.242900	1.17876	17	24.6	8 weeks	male	NaN	NaN
2	16.08.31.1.2	-74.4443	221.895	27.983400	1.17550	7	24.6	8 weeks	male	NaN	NaN
3	16.09.01.0.1	-75.8549	294.484	31.245583	1.10705	29	26.0	8 weeks	female	NaN	NaN
4	16.09.01.0.2	-66.3612	271.674	34.679290	1.06450	25	26.0	8 weeks	female	NaN	NaN

Dep. Variable:	Vrest	R-squared:	0.662
Model:	OLS	Adj. R-squared:	0.408
Method:	Least Squares	F-statistic:	2.611
Date:	Tue, 06 Sep 2016	Prob (F-statistic):	0.188
Time:	22:39:27	Log-Likelihood:	-19.755
No. Observations:	8	AIC:	47.51
Df Residuals:	4	BIC:	47.83
Df Model:	3

	coef	std err	t	P>\|t\|	[95.0% Conf. Int.]
const	-65.9312	21.957	-3.003	0.040	-126.894 -4.969
InputR	-0.1582	0.069	-2.279	0.085	-0.351 0.034
SagRatio	-14.6860	16.502	-0.890	0.424	-60.503 31.131
mbTau	1.6515	0.924	1.788	0.148	-0.913 4.217

Omnibus:	2.275	Durbin-Watson:	2.142
Prob(Omnibus):	0.321	Jarque-Bera (JB):	0.762
Skew:	-0.752	Prob(JB):	0.683
Kurtosis:	2.835	Cond. No.	3.50e+03